All language subtitles for [English (United States)] Statistics - A Full University Course on Data Science Basics [DownSub.com]

Good day,

everyone. This is your lecturer, Monica wahi.
And we're going to start now with section

1.1. What is statistics? So here's our learning
objectives for this lecture. At the end of

this lecture, the students should be able
to state at least one definition of statistics.

Yes, there's more than one, give one example
of a population parameter. And one example

of a sample statistic. Also, the student should
be able to classify a variable into quantitative

or qualitative and as nominal ordinal, interval,
or ratio. So what we're going to cover in

this lecture is, first I'm going to go over
some definitions of statistics. Like I said,

there's more than one. But they all sort of
relate to the basic concept of why you're

doing statistics, and especially not math.
So what's the difference, right, then we're

gonna go over a population parameter and sample
statistic. And you'll know what those mean,

at the end of the lecture. And finally, we're
going to go over classifying levels of measurement.

So let's start with the definition of statistics.
And so we're going to go over these concepts

like what it is. And also I'm going to define
for you the concept of individuals versus

variables. You may know definitions for those
words already, but I'm going to give you them

in statistics ease. And then I'm going to
give you examples of statistics, individuals

and variables in healthcare. So here are the
definitions. What is statistics? statistics

is the study, how to collect, organize, analyze,
and interpret numerical information and data.

Well, that sounds pretty esoteric, right?
But if you actually think about it, even if

he did a simple survey, like you just did

a wiki, you just

look on Yelp, right? You look on Yelp, and
you see, you know, the restaurant, you want

to go to some people say five stars or four
stars, but there's a few two stars one star

will do you go? I mean, there's a whole bunch
of different answers. So how do you do that,

you kind of have to analyze it, you kind of
have to interpret it. So it's not that easy.

So statistics is both the science of uncertainty,
and the technology of extracting information

from data. So in other words, if you've got
a bunch of data about like a restaurant, um,

you don't know how it's gonna be if you actually
go there, right? You don't know for sure.

But, uh, so it's the science of uncertainty.
If you look on Yelp, and you're seeing almost

everybody's giving it a four or five star,
maybe it's gonna be good for you, right? But

you don't know, maybe there's new management.
That's the uncertainty. So statistics is used

to help us make decisions, not just whether
to go to the restaurant or not, but important

statistics, such as in health care and public
health. Well, I guess if it's an expensive

restaurant, maybe it's important. But anyway,
and health care and public health, you really

need these statistics, because they really
guide you. Like, for example, let's think

of the Center for Disease Control and Prevention
in the United States. So what do they do?

They spend the whole year studying the different
flu viruses that go round, because there's

more than one.

They spend

the whole year doing that they organize, analyze,
and interpret numerical information and data

about these different viruses, the different
influenza viruses that are going around. They

extract that information. And you know, what
decisions I make the make the decisions about

what viruses to include, in the next year
sexy? Are they always right? Sure enough,

they're not. I mean, have you ever had a year
where you're like, Oh, my gosh, everybody

I know, got vaccinated, and they're still
getting sick? Well, you know, give him a break.

It's this sign some uncertainty, they it just
didn't work out that time. However, this is

probably better than just randomly guessing.
Right. So that's statistics for you. Know,

I promised you I'd tell you the statistics
ease version of individuals and variables.

Now, if you're outside statistics, you know
that individuals are people, right. And you

know that a variable is a factor, like a factor
that can vary, you know, like, the only variable

is I don't know what time

something's going to happen.

But when you

enter the land of statistics, there are specific
meanings to these two words. Individuals are

people or objects included in a study. So
if you're gonna do an animal study with some

mice in it, those would be the individuals.
If you do a randomized clinical trial, and

you include people who have Alzheimer's in
it, then patients are your individuals. But

we do a lot of different things in healthcare.
We sometimes study hospitals, like the rate

of nosocomial infections, in which case if
you're looking old bunch of stuff in hospitals,

those would be the individuals. Sometimes
we look at states rates of infant mortality,

for example, in different states, in that
case, states would be individuals. So as you

can see at the bottom of the slide, a variable
then is a characteristic of the individual

to be measured, or observed. I give some examples
on the slide. But like I was saying, you know,

if you wanted to study a hospital, for example,
I gave you the example of a variable of a

rate of nosocomial infections, you could also
have other variables about that individual

or hospital,

like the rate of in hospital mortality. And
so, as you can see, one of the things we do

in statistics is we sit down and we decide,
well, who are going to be our individuals

that we're going to measure? And what variables
are we going to measure. So I just threw up

here a few examples of different kinds of
individuals we have, that we use a lot in

health care and public health, and an example
of just one variable, about those example

individuals. But there would theoretically
be many variables about them. And I just want

you to notice, a lot of times, the individuals
are geographic locations. Other times they

might be institutions, like I said, like hospitals,
or clinics, or programs. There's other things

that they are, but these are just kind of
the big ones. So, um, as I was describing,

and just to review, what I went over, statistics
is used in healthcare and other disciplines

to, to aid in decision making, like I gave
the example the CDC and their vaccine for

influenza. And so therefore, it's really important
to understand statistics, because you need

to understand these processes in healthcare,
like how do we figure out

what to do?

Like not only what do we do, but how do we
figure out what to do. And that's really important

because we use statistics a lot in healthcare.
Now, we're going to move on to talk about

what a population parameter is, and what a
sample statistic is. So we're going to go

over first definition of a population and
the definition of a sample. So you're sure

about what those mean. And we're going to
talk about the data about a population and

the data about a sample and how those are
different. And then we're going to get into

what I was just describing parameters and
statistics. And I'll give you a few examples.

So let's start with what is the population,
again, another case where you just have a

normal word, but it has a special meaning
and statistics? Well, it's a group of people

or objects with a common theme. And when every
member of that group is considered this population,

right. So here, here's just one example. So
the theme would be like nurses who work at

Massachusetts, Massachusetts General Hospital,
so the population then if that was your theme,

will be the list from human resources of every
nurse out currently employed at mgh. Now,

it really does depend on how you define that
thing. Like I could have said, nurses who

belong to the American nursing Association,
right? And then we'd be looking at a different

list. I could say nurses who live in New Orleans,
in the city limits of New Orleans who live

there, right, then we'll be looking at a different
population. So really has to do with the details

of how you describe the theme around that
population. But the point is, once you describe

that theme, the population is every single
individual in there. So then, what is the

sample? Well, it's a small portion of that
population. It can be a representative sample,

but it can also be a biased sample, and we're
going to get into that. So let's just go back

to mgh. And think let's say we were going
to survey a sample of the population of nurses

at mgh, let's say we only surveyed nurses
in the intensive care unit. That would be

a sample, but not a representative sample.
So it would be a small portion of that population,

but not a representative one. Probably more
representative would be if we asked at least

one nurse from each department. And so I just
want to get in your head that the whole concept

of sample is, is that it's just a small portion
of the population. And it's not a portion

of some other population. It's just that one.
But the problem is you can get a biased one

or representative one. So you have to think
about So when you think about it, if you've

got a whole population, then you would get
variables about each individual in that population.

And those variables would be your data. But
if you chose samples, that you know, just

a portion will be a lot less work, right?
You'd still have to get variables about those

individuals, but there's way fewer individuals,
so it probably be easier. So in population

data, data from every single individual in
the population is available. And that's called

a census. So I'm, I knew a person who decided
to do a survey of every single professor at

a college. She didn't take just some professors
from each department, she sent the survey

to every single professor. So she did not
use a sample, she used a census. But in sample

data, the data are only available from some
of the individuals in the population. So if

we go back to the researcher I described,
if she had only taken some of the list, the

email list of the professors at that college,
then she would have been serving a sample.

And that's actually very commonly used in
research studies, especially if patients,

why would you need to go get every, for example,
kidney dialysis patient and study every single

one, you only need a sample. And why is that
because we have statistics. So I'm going to

just give you a few examples of real population
data in healthcare. You're probably familiar

with Medicare, Medicare is the public insurance
program in the United States, for elders.

So even my grandma was on Medicare when she
was alive,

and she was not a US citizen, she was from
India. So we really do a good job of covering

our elders in the US with Medicare. In fact,
I even read a statistics that said, almost

100% of people aged 65 and over are in Medicare.
And so therefore, if you download data from

Medicare, they make it confidential, you only
just replace all the personal identifiers.

But there's this thing called the Medicare
claims data set for every single transaction

that happens, like if you're in Medicare,
and you go get some treatment that's in there.

So it has all the insurance claims filed by
the Medicare population, because it has everybody,

everything than that is population data. Also,
in the United States, every 10 years, the

government hires a bunch of people to go out
and survey a bunch of people. And also, they

send out a bunch of surveys. And the idea
is to try to get every single person in the

United States to fill out that survey. And
that's called the United States Census. So

now, I'm going to give you sort of a mirror
image of the sample data. Okay. Remember how

I was just talking to about Medicare? People
who are enrolled in Medicare are called Medicare

beneficiaries, and Medicare cares what they
think. So they do a survey of a sample of

individuals on Medicare. And they do this
kind of often. I think they do it once a year.

I'm not sure it's a phone survey. They only
do a sample because they're going to use statistics

to try and extrapolate that knowledge back
to the population of Medicare beneficiaries.

Also, in case you notice, the United States
Census only takes place every 10 years. Do

you think changes happen in between? Yep,
lots of changes. Like you just think about

Hurricane Katrina. That's very sad. It changed
the population distribution in Louisiana,

vary vary dramatically, and also other states
around there. So how did they keep up? Well,

they used the American Community Survey, the
government does this the United States Census

Bureau, and that, again, is done by phone.
And that's conducted yearly. And it's a sample

and so the US doesn't know exactly how many
people would be in Louisiana or anywhere else.

But they can use statistics to extrapolate
that from the sample of the American Community

Survey. I want to just do a shout out to statistical
notation. So from now on, when we see a capital

N, like let's say you sack capital N equals
25, then you can assume that 25 means a population

that's just kind of a secret code we use in
statistics. However, if you saw a lowercase

n, n equals 25, and it was lowercase, then
you could assume that this was a sample of

the population. And again, it's just kind
of like a secret code, you have to pay attention.

When I'm talking and I say n, and you can
see uppercase and lowercase. You don't know

if I'm talking about a population, or a sample.
Now I'm going to get into the concept of parameter

versus statistic, I want you to notice that
the word parameter starts with P PA. So parameter

is a measure that describes the entire population.
So for instance, anything that would come

out of that whole Medicare claims data set,
or that whole United States Census would be

a parameter. On the other hand, a statistic
statistic starts with S, and statistic is

a measure that describes only a sample of
a population. Here we have an, again, a situation

where the word statistic is used, like daily
on the news. In fact, sometimes I hear on

the news, something like Oh, look at the rate
of HIV in Africa, it's going up. That's a

terrible statistic. I agree. It's terrible.
But they mean parameter, because they're talking

about all of Africa, every single person in
Africa, if the rate of HIV is going up in

Africa, they mean a parameter, they don't
need a statistic.

So here's an example of parameters and statistics
that are based on the same population. So

for example, the mean age of every American
on Medicare is a parameter that's every single

person. However, remember, the Medicare beneficiary
survey, that's just a sample. So if we took

the mean age of those people, we would just
have a statistic. And again, you just have

to pay attention, because if you listen to
the news, you'll hear them use the word statistic

to mean both parameter and statistic. But
in this situation with, when you're practicing

in the field of statistics, it's very important
to point out when the number you're talking

about comes from a population versus comes
from a sample. So you should really use the

term. This is a parameter if it's from a population,
or this is a statistic, if it's from a sample.

And so again, don't get confused. If you're
listening to someone talk in a lecture or

in a video, you might want to look for clues
that a number is a population parameter, or

as a sample statistic, if you hear that the
data set that they use encompasses an entire

population. And usually that's the kind of
stuff done by governments, like remember when

I was talking about the rate of HIV in Africa,
lead probably be done by governments of the

United Nations, or the World Health Organization.
So when you're talking about numbers that

might have come

out of an

entire population, usually done by the government,
that's probably a population parameter. clues

that someone's talking about a sample statistic
is if you hear them talking about a study

that recruited volunteers,

well,

then, if it's volunteers, they didn't get
everybody in the population. So it's going

to be a sample. Also, like surveys, for instance,
surveys about who people are going to vote

for you public opinion surveys, they're never
going to ask some every single person in the

state, who are you going to vote for build
us ask a sample. So if you hear about a survey,

you might even have them tell you say, n equals
maybe a few 1000 people because that's all

they surveyed. And so that's a clue that we're
talking about a sample statistic rather than

a population parameter. Now, I'm going to
talk about the difference between descriptive

statistics and inferential statistics. But
first I'm going to remind you what the word

infer means. So infer means to kind of get
a hint from something indirectly. It's kind

of the complement to imply. So if I said my
friend implied that I should not call after

9pm and I figured that out. I would say I
inferred that I should not call my friend

after 9pm. Okay. So in inferential is what
I'm going to talk about next. But first I'm

going to talk about descriptive descriptives
is pretty easy, because you can do it to samples

and you can do it to populations will variables
from samples and populations, right. And so,

descriptive statistics involve methods of
organizing picturing in some Rising information

from samples and populations. It's basically
just making pictures of it right? Like look

at that bar chart. And that's just a simple
picture. And that can be made with just about

any data. You get data from surveying people
at work, you get data from surveying your

friends, what they're going to bring to the
potluck. If any of that can be used, you can

go download the census data, you can make
descriptive statistics out of that. But there's

something very special about inferential statistics.
And that involves methods of using information

from a sample to draw conclusions regarding
the population. Therefore, inferential statistics

can only be done on a sample. And therefore
and that's why that's called inferential.

Right? Because infer, because the sample is
going to give a hint about what the population

is right? It's not going to say it directly,
which is annoying, right? But that's that

uncertainty thing I was telling you about.
So the sample is going to imply something?

Well, we're gonna infer something from the
sample about the population, right? So that's

what inferential statistics is, is where you
take a sample, and you infer something about

the population. Whereas descriptive statistics
is more loosey goosey. You can just do that

to samples and populations, kind of like make
pictures out of it, right. So in statistics,

it's really important to properly identify
measures as either population parameters,

or sample statistics. Because as you can see,
you can only do inferential statistics on

samples. And so you have to really know what
you're doing when you're doing statistics,

what you're talking about, because different
types of data are used for parameters versus

statistics. Alrighty, now we're going to get
into classifying variables into different

levels of measurement. So remember our variables,
right, like we have individuals, and then

we have variables about them. And those variables
actually can only fall into two groups, quantitative

versus qualitative. And then depending on
which group they fall into, you can further

classify them as interval versus ratio, or
nominal versus ordinal. And I'm going to give

you some examples of how to classify a few
healthcare data, types of variables already,

so I like to draw this picture. It's a four
level data classification, I'll draw it solely

here for you. So we start with human research
data, that's what I like to start with. Alright,

so we're going to split that into two. Remember,
I said that, we're going to start by talking

about quantitative. Another word that's often
used for that is continuous, but we're going

to use the word quantitative. So what does
that mean? That is a numerical measurement

of something. So like, this gives an example
of temperature. So something

with a number in it, I always think if I can
make a mean out of it, it must be a quantitative

variable, right? And so here's an example
of quantitative variables. So time of admin,

right? So imagine that you work a shift in
the ER, right? And from maybe 8pm to 12. like

midnight, right? So you have this for hours.
And you could say, what the average time of

admin would be for those who got admitted
to the hospital, you know, somebody got admitted

at like, eight o'clock, and then somebody
at 815, and whatever, you could put that together,

and you'd say what the average time was, also,
like, if you were doing a study, and you as

you were saying, patients with a particular
condition like Alzheimer's disease, you could

ask them their year of diagnosis, and then
you could make an average of that. And so

you know, that that is quantitative. systolic
blood pressure is also numerical, and platelet

count. And these are variables we run into
all the time in healthcare. So we're, you

said that this is quantitative. Now, we'll
get back to our picture. So that's one side.

So what if it's not quantitative? What else
could it be? Well, the only other category,

it could be is categorical or qualitative.
I use the term qualitative, but some people

use the term categorical, but that's kind
of what it is, is that it's a quality of something

or a characteristic of something like sex
or race. So here are some qualitative variables

in healthcare, like you can have type of health
insurance, like whether you're on Medicare

or Medicaid or different types of private
insurance. Those are all just categorical,

right? You can't make a mean out of that.
Also country of origin. If you're in our group

of students and their international students
in there. Well, what countries are they from?

Right? Well, you can't make a mean out of
that. Also you have situations where you do

have numbers involved, like the stage of cancer,
right? That's depressing. Stage One, cancer,

stage two, cancer, stage three, well, you
never can make a mean, out of the stage of

cancer, you wouldn't say, well, the mean stages
is 1.4, or something like that. It's just

a category. And of course, stage four is a
lot worse than stage one. You know, they're

not just equal categories, but their categories.
Same with trauma center level level four Trauma

Center, where you wouldn't make a mean out
of the number of after the term Trauma Center,

right, like what level it is. But you could
say, well, in the state, maybe. So many percent

of our trauma centers are level four trauma
center. So it's really just a categorical

variable, even though there's a number involved.
Alright, so let's get back to our diagram,

we figured out how to take any variable, and
first split it into one of two categories

is either quantitative, if it's numerical,
or qualitative, if it's a characteristic.

Now, we're going to just concentrate on quantitative
because we're going to separate those variables

into two categories. And the first one we're
going to look at is interval. And the second

one we're going to look at is ratio. So if
a if you happen to decide a variable as quantitative,

then it could be interval or ratio, but not
if it's qualitative. Okay, if it's qualitative,

it doesn't get to do that. So let's look at
interval versus ratio. So on the left side

of the side, we have interval, which is where
it's quantitative, and the differences between

data values are meaningful.

And

ratio has the same thing, the differences
between the data values are meaningful. What

does that mean by that? Well, remember how
I was talking before how level one trauma

center and level two trauma center that that
those are really categories, and not quantitative

variables, because the difference actually
between them is not equal. Especially if you

think of job classifications that might go
in 1234, like nurse, one, nurse to nurse three,

nurse four, or I worked at a job where we
had office specialist one, office specialist

to Office specialist three.

And you know what the deal

for going from office specialists to to Office
specialist three was really hard, you really

had to do a lot there. But to go from one
to two wasn't that hard? So that was a categorical

variable, right? Because the differences between
the values were meaningless. Okay. Like the

difference between s one and s two versus
Oh, s two, and s three, they weren't equal.

Whereas when you're dealing with a quantitative
variable, regardless of whether it's interval

or ratio, you're talking like years, or systolic
blood pressure, one year for you is one year

for me. So that's fine, right? But here's
where the difference comes in between interval

and ratio. So all quantitative variables have
meaningful differences between their data

values, but this hairsplitting thing here
is that an interval, there is no true zero.

And in ratio, there is a true zero. And this
is how I try to think about it. an interval

means kind of like, a space between two things.
Like if you think of the word intermission

is kind of like an interval. It's like an
interval of time during a show where you get

to get up and go the bathroom and get some
coffee. So that's interval. And so if you

have something that's a space in between,
that's not going to have a zero, it doesn't

really start anywhere, or end anywhere. It's
in between. Whereas ratio, how are you number

that is, I don't know if you remember from
like high school, but you can't have a zero

on the bottom of a ratio or a fraction. So
that's the way I use a pneumonic. That ratio

means that you cannot have a true zero. But
how does this work out literally? Well, I'll

show you. So let's go back to those examples
I showed you of quantitative variables, right?

Because those are the only ones we have to
make this decision about whether they are

interval ratio. So these are these examples.
Now I'm going to remind you that ratio has

a true zero. Remember that little pneumonic
I said, like don't divide by zero. And so

you know, like in a ratio, so they have a
true zero. Well, let's think about it. It's

not very pleasant to have a zero systolic
blood pressure because you'd be dead. Same

with the platelet count, but it is possible,
right? But now when we go on to interval,

we can't have Like zero time, like time of
admet, you know are your diagnosis, there's

no like, year zero. So as you probably just
guessed, ratio is where it's at. In healthcare.

There's not a whole lot of times when we have
interval data, but we do, you know, anytime

you have a time, so you got to keep that in
mind that if you want to split your quantitative

variables into either interval or ratio, you
got to keep this in mind the difference between

the true zero and the no true zero. Okay,
here's our handy dandy diagram. We've just

gone through the tree classifying quantitative
data into interval versus ratio. Now let's

go pay attention to the other side of the
tree qualitative. So how do we split those?

Um, well, we can split those into nominal
versus ordinal. All right. So nominal applies

to categories, labels, or names that cannot
be ordered from smallest to largest. Okay,

like I kind of think of when they have an
advertisement, they say, for a nominal fee,

you can do this, it means it's small, they're
like, there's almost no difference. And so

that's why I say, there's no difference, it's
not smallest to largest is means they must

be equal. That's how I remember it in my mind.
But then ordinal applies to data that can

be arranged in order in categories. But remember
that thing I was saying about quantitative,

it's not quantitative, right? Because the
difference between the data values either

cannot be determined or is meaningless, like
I was talking about with cancer, especially,

you know, if you go from stage three to stage
four, that's materially different than stage

one to stage two. So you really can't determine
those things. So this is where we're gonna

get into that it's ordinal. It's arranged
in categories that can be ordered from smallest

to largest. So remember, our old friends that
I threw up there before of these examples

of qualitative variables and healthcare? Well,
let's just reflect on this nominal cannot

be ordered, right. So that would be more like
type of health insurance and country of origin

because they could all be equal. Whereas ordinal
is going to have a natural order, even though

the differences between the levels is meaningless,
which is what makes it so different from a

quantitative variables. So which is why it
stays on the qualitative side of the tree,

it just gets labeled ordinal. So what you
want to do is if you think you have a qualitative

variable on your hands, look for a natural
order. If there is one, it's ordinal. And

if not, it's nominal. So all data can be classified
as quantitative or qualitative. So if you

have a variable, that's the first split you
can make as the difference between quantitative

and qualitative, but once you do that, you
can further classify it as interval ratio,

nominal, or ordinal. And it's really important
to know how to classify data in healthcare,

as you'll find out later. Because depending
on how you classify it, you might be able

to do different things with it in statistics
already, so what we went over was the definition

of statistics. And we talked a little about
why you use it and how you use it, especially

in healthcare. We went over what it means
to talk about a population parameter and the

sample statistic, and we went over some examples
about them. And then we talked about classifying

variables into the different levels of measurement,
and even talked about a few examples there.

So I hope you enjoyed my lecture. Greetings,
this is Monica wahi lecturer at library college,

bringing you your lecture on section 1.2 on
the topic of sampling.

So here

are your learning objectives for this particular
lecture. At the end of this lecture, the students

should be able to define sampling frame and
sampling error, the student should be also

able to give one example of how to do simple
random sampling. And one example of how to

do systematic sampling. The students should
be able to explain one reason to choose stratified

sampling over other approaches, state to differences
between cluster sampling and convenience sampling,

and give an example of a national survey that
uses multistage sampling. So let's jump right

into it here. So we're going to go over in
this lecture, sampling definitions, and then

those different types of sampling I mentioned
in the learning objectives, simple random

sampling, stratified sampling, systematic
sampling, and then convenience and multi state

sing. So let's start with some sampling definitions.
What is a sample Okay, so we're going to revisit

that concept from the previous lecture, we're
also going to talk about sampling frames,

and what errors mean and errors of sampling
frames. And then we're also going to just

go right back over that and make sure you
understand before we go on, and talk about

the different types of sampling. So we take
a sample of a population, because we want

to do inferential statistics, remember that
we want to infer from the sample to the population.

And it's just not necessary to measure the
whole population, it would be impractical.

And it's cost a lot. And actually, what you'll
find is, if you ever do an experiment, when

where you actually do measure the whole population,
you'll find that if you get, you know, a pretty

good proportion of the population, and you
just take that, you, that's all you really

needed to talk to. So ultimately, we save
resources, especially in health care, when

we do a good job of sampling, and use that
to infer to the population rather than having

to take a census of the whole population all
the top. So that brings us to the concept

of sampling frame. So the sampling frame is
the list of individuals from which a sample

is actually selected. And the list may be
this physical concrete list, like you could

have a list of students enrolled at a nursing
college, or in my other lecture, I gave an

example of a list of nurses who work at Massachusetts
General Hospital, that could be your list,

you'd go to human resources and get that.
Or it could be a theoretical list. It could

be like the list of patients who present to
the emergency department today, obviously,

when you go into work, at the beginning of
the shift, you're not going to know who's

on that list yet. But it could be a theoretical
list. But whatever that list is, that is your

sampling frame. So that those are the people
who actually could be selected for your study.

So the sampling frame is the part of the population
from which you want to draw the sample. And

you want to work at such that everybody from
your sampling frame has a chance of being

selected for your sample. In other words,
you don't want to leave anyone that should

be in your sampling frame out in the cold.
That leads us to the concept of under coverage.

So what is it? It's omitting population members
from the sampling frame? They're supposed

to be on the list, but they're not there.
So how can this happen? Well, let's say you

did what I was suggesting in the previous
slide, you got a list of nursing students,

you know, from a college, let's say somebody
signed up that day, or somebody was just admitted

that day, maybe they didn't make it into the
database in time and you're missing them.

Or even like that HR list I talked about,
at mgh, well, you know, I know how nurses

are, sometimes they'll temp in different places,
and maybe they're not on the payroll, maybe

they're through a temp agency. And so then
we would miss those nurses from the sampling

frame. And then, you know, people who present
at the emergency department at night might

be different than those in the day. And so
if you're really trying to sample from people

who present to the emergency department, you
can't just look at like some small period

of time, you'd have to look at, you know,
the whole 24 hour cycle. So if you omit population

members from your sampling frame, they don't
even get a chance to be in it. And that's

called under coverage. Now, I'm going to shift
around, we're jumping around with a few different

definitions.

And we're going to talk about errors. Now,
this is something that took me a while to

get used to in statistics, there's actually
two kinds of errors in statistics. The first

kind is I call it This is my own terminology,
a fact of life error. It's just an error that

happens. When you do statistics, it's not
bad or good. It's just what happens. And in

this case, I'm going to describe one of those.
It's called a sampling error. So the sampling

error just simply says the population mean
will be different from your sample mean, and

the population percentage will be different
from your sample percentage. So what does

that mean? That means that if I cut corners,
like I said, I could write and just take a

sample to infer to the population. If I actually
do one of those experiments I was telling

you about where I have the population data
and I just take a sample and compare the means

they will be different. Okay, I mean, there
might be this huge coincidence where they're

the same but they're typically different.
Same if you do percentages, and and we just

know this is going to happen. The statistics
we account for it, we have ways of dealing

with it. But we know that there's always going
to be sampling error whenever you take a sample

from a population To try to make a mean or
percentage in the sample, it's just not going

to be exactly what's in the populations fine.

But then

there are other errors and statistics, which
are actually bad. And your it means you made

a mistake. It's like mistakes, literally mistakes.
And so as you go through learning about statistics,

it's almost like you have to sit down and
ask somebody, is this one of those fact of

life errors? Or is this one of those errors
you want to avoid? Well, we just talked about

sampling error. That's just a fact of life
error. But errors, you want to avoid non sampling

error. That's basically using a bad list.
I had an example in my life where I wanted

to study a whole bunch of providers, right.
And my friend gave me this list of providers,

and and said, this is the entire list of all
these providers in this particular professional

society. But when I sent the email to that
list, I found there were not only duplicates

on this list, but a lot of people emailed
me back and said, Why are you sending this

to me? I'm not a provider. I'm not part of
this professional society. And also, some

people who were in that professional society,
who had heard about the survey emailed me

and said, Why didn't I get the survey. So
this was a bad list. Some people had been

left out of the sampling frame. So people
who were in the society somehow weren't on

my email list. And that's a problem, right?
So you have to pay careful attention. This

was actually a mistake I made, you have to
pay careful attention that everyone in the

population who was supposed to be represented
in your sampling frame is actually there.

So I should have really done a better job
of calling the professional society and making

sure that this list was a good list. So sampling
error was caused by the fact that regardless

of what you do, your sample will not perfectly
resent represent the population. Whereas non

sampling error, yeah, I was sloppy. It was
poor sample design, sloppy data collection,

and accurate measurement instruments, you
can have bias and data collection, other problems

introduced by the researcher. So this is your
fault if there's non sampling error, but sampling

error is just a

fact of life.

Little whiplash here, we're gonna now move
on to the concept of simulations. So a simulation

is defined technically as a numerical facsimile,
or representation of a real world phenomenon.

So it's like working through a pretend situation,
to see how it would come out in the case that

was real. And this, you know, when you study
statistics, you end up doing a lot of simulations.

And remember how I've been talking about an
experiment you could do if you somehow did

a census and had a whole bunch of data on
a population, you could do an experiment where

you just took a sample from that population
and looked at their mean to see the sampling

error. That's an example of a simulation.
So to just conclude this little section, it's

really important to do your best to avoid
non sampling error. And this is achieved by

making sure you do not have under coverage
when sampling from your sampling frame. So

this puts together some of our vocabulary.
But just remember, sampling error is a fact

of life. Okay, now we're going to specifically
talk about different types of sampling. And

we're going to start with simple random sample.
Okay, so first, we're gonna start with just

explaining what is meant by simple random
sampling, then we're going to talk about two

different methods of doing simple random sampling,
they work the same way they achieve the same

thing. It's just that depending on how you're
doing your research, one might be more convenient

for you than the other. Finally, we will go
over the limits of simple random sampling,

because all these sampling methods seem perfect.
But then you got to take a look at their limitations.

So let's first define simple random sampling.
So here's a definition. A simple random sample

of n measurements from a population is a subset
of the population selected in such a manner

that every sample of size n from the population
has an equal chance of being selected. Well,

it's kind of complicated, but what it means
is, is that if you use the proper approach

for simple random sampling, whatever sample
you get, you could have had just as easily

a chance of getting another batch, another
group of people from that sample. In other

words, like, let's say you have a list of
the population of students in the class. So

I'm going to define a class as a population.
And you want to take a sample of five students

from this bigger class. If you take a simple
random sample, it means that all the different

groups of five students you could pick from
the list has an equal chance of being the

sample group you actually pick. Now, you can
just imagine that if you race into the class

right at the beginning, and you take your
sample of five and not everybody's in the

class, what does that sound like, right, a
sampling frame problem, maybe an under coverage

problem, maybe biases creeping in there, right.
And so you just got to be careful, if you're

going to do simple random sampling, that you
start with a list with everybody in your sample

frame, because every single sample that you
could possibly take should have equal chance

of ending up being your sample. And I'll kind
of explain it by explaining the two different

methods that can be used of obtaining that

sample.

So one of the best things that you can do
is just start with a really good list of all

the people in your population. So maybe, you
know, if I was going to study, I used to work

at the army. So let's say I was going to study
all the people who are active duty in the

US Army, I would like to get a list of all
of those people from an accurate place at

the army. And I would like to have them have
a unique ID. Okay. And that's true in the

army, everybody in the army has a unique numerical
ID. So what I would do, like in here, if you

were looking at students, you'd take maybe
take a student ID, so then you take the IDS

from everybody on the list, and you cut them
up, like you print them out, and you cut them

up, and you put them in a hat, right, or a
bag where you can't see in it. And they mix

them all up where you can't see it. And you
draw five of them up, or like in the picture,

you know, what they did was mix up all those
papers, and now they're not looking. And they're

drawing a few out. Okay, so what did you just
do, you just made sure, first of all, that

everybody in the population had an ID number.
And that when you printed it out and cut it

up, all, you didn't lose any of them, if you
drop them on the floor, or something that's

not simple random sample, you got to make
sure you keep all of them, and that you put

them all in the hat, and that you didn't look
and you draw five or whatever, because then

any five of those slips of paper could have
been drawn in there for your meeting with

simple random sampling. Okay, that method
will work, right? Another method that works,

that might work better if you can't do this
ID thing where you cut a paper is where you

simply just make your own list of unique random
numbers, right, you just make your own list.

And then you assign those to the population.
A great example is if you're, you know, kind

of teaching kids and you want to put them
in a random order, maybe you're gonna do a

game or something. Well, all you do is you
you get, like, let's say you have 10, kids,

you number one to 10, you put it in the hat,
and then you pull out the first number, let's

say it's five, you give it to the first kid,
right? And then you just keep pulling out

numbers and giving them to the kids and then
tell them to stand in order, right? So you

generate a list of random numbers as long
as the list of the population. So I said,

What if you have 10 kids? Well, if you have,
you know, 500 names, then you get 500 numbers,

and they don't have to be one through 500.
They just have to be unique. Okay, I like

smaller numbers. So I'd say keep them small,
but you can do what you want. And then, in

any case, you randomly assign these numbers,
you can use the hat, I'm big on hats to this

population. And then, you know, you ask them
to stand in order, or somehow you figure out

it's kind of like a raffle you call out who's
got number one, you know, and whoever says

yes, you're like, you're lucky you get to
be in my study, you know, so you can take

the first five numbers in the order, right.
And that's, that'll achieve the same thing

as the last method, you'll get a simple random
sample, it's just two different ways of doing

it. So ultimately, being in a simple random
sample means that the sample has an equal

chase chance of being selected out of the
hat that this group of people or a group of

whatever has an equal chance of being selected.
And you'll see this picture on the left here

is bingo, as some of you may play bingo. You
know, they pull balls out of there and they

call off the names of the balls. Well, each
ball has a unique actually a letter and a

number unique on there. And that's how they
make them random. That's they take a simple

random sample of these bingo balls each time
that they do a bingo game. So I described

to you the first method of doing that using
an old fashioned hat. The second method, you

know, where you generate your own numbers,
and you just make sure they're unique. And

then you assign them to things and put them
in order. Well, that's my electronic hat.

That's how I handle it. If I have, for example,
somebody sends me an Excel sheet with a list

of hospitals on it. I'll just assign each
hospital random number and sort them in order.

And I'll sample the top few hospitals. That'll
be how I get a simple random sample of possibles.

That way, I'm not biased, picking out my favorite
hospitals where all my friends work, right?

If I do it that way, the first method or the
second method, all members of the population

have the equal probability of being selected
in the sample. And more importantly, all possible

samples, all possible groups had an equal
chance of being selected. Of course, I only

did it once. So I only got one of them. But
the other ones that weren't selected had an

equal chance of being selected.

All right, you probably saw the limits, is
this whole list? Even if I'm sampling hospitals,

right? I still need a list of hospitals to
sample from. So you may not know who's gonna

show up in the emergency department that day,
if you do, while you're psychic, because most

people are not. So how would you sample from
them using simple random sampling? So simple

random sampling is okay, when you got a list
like hospitals, but it's not so good when

you don't know who's going to show up that
day. And even if you do a simple random sampling,

you need a good list. I made a mistake once,
where I did a survey with a bunch of professionals

using a professional society list. And when
I sent out the survey, I learned that there

were people on the list who were no longer
part of the society that it was an old list.

And more importantly, there were people who
had joined the society that had not made it

onto that list. So I was getting under coverage.
So like, if you were doing a study with students,

you know, what if they just left off the part
time students, then you'd be missing them.

So this is a great example of non sampling
error. And so if you're going to do simple

random sampling, you do need a list and you
really want to research it and make sure it's

the best list possible. So I just went over
the characteristics of simple random sampling,

and two different methods you can use from
to sample from a list. And I also mentioned

the limits of it. Now we'll talk about a different
kind of sampling, stratified sampling. So

we're gonna go over what it is. And then I'm
just like, simple random sampling had all

these steps to it, there are different steps
in stratified sampling. And I'll give you

some examples. And then of course, just like
simple random sampling, this stratified sampling

has limitations, and I'll talk about those.
So I first wanted to just remind you what

the word stratified means, or what strata
are, the single word is stratum, and more

than one a strata. Now you see that rock on
the slide, you see that big, horizontal line

across it, that those that's a stratum, there
are strata, right? Those are strata of rock,

if you stay geology, that'll the geologists
will explain that where those breaks are,

it means something happened often in the weather
or the environment or whatever. But the reason

why I put this picture up there is I want
you to sort of imagine those layers. Because

that's what we do in stratified sampling is
first, we divide our list, of course, you

know, a list, we divide our list into layers.
Okay, so remember how I was just talking about

simple random sampling? Like, what if I sample
from hospitals? Well, I could take this hospital

list and divide it until layers by for example,
how close they are to the city, I could say,

urban, suburban, and rural, I could first
put them into those strata. Okay. And if I

was doing that, I'd be doing stratified sampling.
Same with students, like I could put them

in, you know, first year nursing students,
second year students, you know, and I'd have

this them divided into strata first. Um, so
this is what so why would you do that? Why

not just do simple random sampling? Well,
if you think about it, let's say that you've

got a class like statistics, maybe a lot of
you know, they're not that many first year

students in it. So let's say the very small
proportion is that way. If you do simple random

sampling, you might just by lock miss all
of them. Right. And so, if you're really concerned

about what a minority thinks, then you can
make sure to get representative from that

stratum. By doing stratified sampling, because
the first thing you do is you put those that

list into groups. And then you take a simple
random sample from each of the strata. So

here's the steps. So step one, divide the
entire population, the whole list you have

into distinct subgroups called strata. And
remember, each individual has to fit into

one of those categories. So if you have somebody
who's sort of halfway halfway between first

year and second year, or you've got a hospital
that's kind of on the border, it you got to

choose, you got to put it in one of those
groups. Step two, um, well, it's not really

step two, but you've got to think about the
strata like what is it based on, it's got

to be based on one specific characteristics,
such as age income, education level, you know,

a great example is you could take people of
all different incomes, right, that's a quantitative

variable, but you can put them in strata by
you know, less than a certain amount. And

then that to that, that to that you can make,

you know, four or five strata. And then, um,
you know, you just want to make sure that

all members of the stratum, each stratum,
share the same characteristic. And then you

could do step four, which is draw a simple
random sample from each stratum. So like,

in the case where I was describing, like,
maybe you have a class with very few first

year students, if you take a random sample
of five from each strata, you know, each stratum,

then you might be, you know, you're kind of
getting almost like, extra votes from a small

minority, right? Like, you're kind of treating
them fairly, even though there's a way bigger

group of the other people you're taking exactly
five from. And, but you just that, that's

the risk you take, because you want to make
sure you hear from that small group. Because

if you just do sample random sampling with
groups, so small, you might just accidentally

miss it. So here are some examples of stratified
sampling. And you'll see this in the youth

Behavioral Risk Factor Surveillance surveys
that they do in high schools, that they'll

stratify by grade, right, because if they
did a simple random sample, you know, a lot

of students drop out of junior and senior
year, they get probably too many freshmen

and sophomores. And so they're gonna want
to look at getting a certain amount of freshman

classes, certain amount of sophomore classes,
certain amount of junior classes, student

run the senior classes, so they can have enough
of each to make good estimates, right. And

in hospitals, they often sample providers
from each department, right? Like, they don't

just do a simple random sample of providers,
if they're asking about like provider satisfaction,

or if you know about a policy, they won't
just do that, because they might, for example,

Miss everybody in the ICU. Or if you're studying,
you know, ICU is you have multiple ICU is

there,

then

you would want to maybe stratify by ICU, just
to make sure even if one of them's smaller,

just to make sure you have

a good,

good solid representation from each ICU. So
those are the reasons that push you to do

stratified sampling. It's not always necessary.
But when you have these situations where you

have these distinct groups, especially the
little one involved, and you want to hear

from everybody, you really want to consider
the stratified sampling. So of course, there's

limitations. And I've been sort of leading
up to this, what you end up doing is over

sampling, one of the groups usually, you know,
like the smallest group, if you make the same

amount of people you take from that stratum,
the same amount as you take from the big stratum.

It's like the smallest group is having all
these powerful votes and the biggest group

has is weaker, you know, they're both equal
when they're not technically equal in the

population. But that's the way it goes, right?
And I do higher level statistics, there's

ways to adjust back for that, to just sort
of say, take a penalty for that and go back

and say, Well, what if the real pot you know,
we can extrapolate this back to the population

proportions? It's possible, but it's it takes
some post processing is just the issue. And

it's also like simple random sampling not
really possible to do without a list beforehand.

And it's also hard to do, because you actually
have to split the list into groups into these

strata. So let's say I had these hospitals
and I didn't know where they were, I didn't

know exactly if they were urban or rural or
suburban. Well, that adds another level of

complexity to this whole stratified sampling.
So, in summary, I just went over what stratified

means, and it means you know, putting things
in groups and then taking from that, and I

describe the steps involved. And it's a stratified
sample. It goes a lot easily. A lot more easily

if the strategist happened to be equal to
begin with, you know, I gave the example of

high schools, usually there's maybe slightly
fewer people in junior and senior year, but

it's kind of close. And it's always nice.
Like if you're comparing ice use, for example,

if the ice use are roughly the same size,
because then you don't have to worry about

this whole, one of them is smaller, but it's
getting an equal vote. Already, now we are

going to move on to talk about systematic
sampling. Okay, well, systematic sampling

actually can be done with or without a list.
So it's a little more flexible than the kind

of sampling we've been talking about. systematic
sampling, it's easier for me to like, define

it by describing the steps you go through
to do it. So I'm just gonna explain how to

do it. And then you'll understand, in fact,
you'll understand why it's called systematic.

So whether you have a list or not, what you
have to do for step one is arrange all the

individuals of the population in a particular
order. Now, if it's a list, you just make

it in whatever order you want to make it in.
But if we're talking about, for example, patients

coming into the ER, well, they come in, in
the order that they want

to.

So they already are arranged in the list,
right? You just don't know what that list

is. Okay, then step two is pick a random individual
as a start. So let's say I had a list of hospitals,

and let's say it was just sorted by state,
right? I, let's say I picked a random individual,

maybe I went down, you know, seven on the
list, and I picked that hospital. Or maybe

you could be at the ER, you start your shift.
And the seventh patient who is admitted to

the ER, you pick that person, just I picked
seven, I mean, you could have picked five,

you could have picked 20, you know, just you
pick a random person. Then the next step,

step three is take every case member of the
population in the sample. Now, don't try this

in Scrabble case is not a word in Scrabble,
okay? It's just a word and statistics ease,

in what case means spelled k th, it means
every so many. So let's pick a number and

fill it in for K. So let's pick the number
three. So let's say after you pick your first

hospital from the list, or the first patient
from the ER, it doesn't matter what number

you chose for that, then you take every third
after that. So every third patient that comes

in after that, you ask them if they want to
be in a study, or every third hospital after

that original random one, I pick and I say,
Okay, this is going to be part of my systematic

sample. So as you can see, it's like pretty
simple to do, it's easy to do, if you have

a list, it's easy to if you don't have a list,
it's just the deal is you have to pick K,

well, first you pick a random place to start,
then you pick K, and then you just keep going

every so many. So you could do this with classes,
you could take out a list of classes available

at your college next semester, she pick a
random number like three, you know, and it's

sorted some way. So you go to the third class
and you circle that, then you pick another

random number like five and then after that
you pick every fifth class. So after the third

one, you go 45678, and then 910 11 1213. And
you keep picking classes. Okay, this is not

career advice. Okay? Do not pick your classes
that way. This was just an example. Alright,

so as you probably guessed, I'm going to be
negative Nelly, again, there are problems

with systematic sampling. If already things
are set up, boy, girl, boy, girl, for example.

If you pick like an even number, you're going
to get all boys are all girls, right? And

I noticed this actually, when I was doing
a study in the lab, we wanted to study like

whenever they put the assay through the machines,
we thought some of the assays weren't running,

right. And so we wanted to take a sample.
And I wanted to take a systematic sample.

But I wanted to take a systematic sample,
like every seven days, and that's a week.

And so I asked my colleague, does the lab
vary day by day in what assez it runs because

of it always runs the sexually transmitted
disease assays, it saves them up and runs

them all on Friday. And I'm sampling from
every Friday, that's all I'm gonna get, right?

That's actually called periodicity. You don't
have to remember that I don't think I've ever

even seen that written. It's just I remember
my lecture in my class telling us that that's

what you have to worry about with systematic
sampling. It's not real common problem, though.

But what's awesome about it is you can do
it in a clinical setting. So you You can sample

patients that way, coming into a clinic or
coming to a central lab or like in the emergency

room. And that's why this is a particular
power, particularly powerful way to sample

is that if you have an ongoing sort of patient
influx, when you design your research, you

could simply say, once you decide how many
people you need to recruit for your sample,

that you would use systematic sampling, and
just have somebody in the clinic inviting

every case person who qualifies every case
patient who qualifies into your study. So

it's easy to do systematic sampling, it's
easy to do with or without a list. And you

just pick a random starting point, and then
you pick every case individual. Next, we're

gonna move on to cluster sampling. So what
is up with cluster sampling? Why do we need

even other kinds of sampling? I just went
over so many kinds. I mean, you could use

stratified systematic or simple random sampling,
why would you even need another kind? Well,

cluster is very special. It's special, because
it's the kind of sampling you use when you

think there's a problem at a particular geographic
location. Typically, that's how cluster sampling

is used. And, and I'll explain it further.

Imagine, for example, there's a particular
factory that's is believed to admit fumes

that cause problems with people's health.
Well, you can't do simple random sampling

all over the nation, right, or you won't even
get people by that factory, can't really do

easily do stratified or systematic sampling
their cluster sampling is what's designed

when you want to study something that's coming
from a geographic location. So when you do

cluster sampling, you start by dividing a
map into geographic areas. So I'm from Minnesota,

and I know that there was a mine there with
vermiculite in it. And it was it was contaminated,

a lot of people got sick from it. But they
didn't know that's what was going on. So they

first I think divided Minnesota into different
geographic areas, areas. after dividing the

area into these different geographic areas,
some with the, with the bad thing in it, and

some without the bad thing in it, you randomly
pick these clusters or areas from the map.

So the app, like if you'll see there on the
screen, there's a map of the state of Virginia,

and it's all been divided into different groups.
And then this, this cluster is is highlighted,

you usually probably pick more than one cluster,
sometimes it's only four or five. But the

idea is you try to enroll all of the individuals
in the cluster, it's usually people, although

you can do it with animals, if there's a disease
going around among animals, you know, you

would have these, the divide the area up into
clusters, and then you try to measure all

the animals in the cluster. So as you can
imagine, not only is this sort of practically

difficult, but there's reasons why people
live together, right? People live in communities.

I mean, people don't just randomly scattered
themselves, you know, cultural communities

grow. companies grow around art, you know,
affluent communities have different people

in them, then communities that have less money.
So sometimes the people located in the cluster

are all similar in a way that makes the problem
hard to study. And this is, especially if

you're studying some geographic thing, like
maybe a factory or a sewage plant, that you

think might be causing cancer, if you're in
an area where there's a lot of pollution anyway,

from other things, and a lot of low income
people live there. Because if you're high

income you can afford not to, well, they're
already being exposed to higher rates of carcinogens

and probably have a higher cancer rate. It's
hard to tell what the independent effect might

be of that thing in that geographic location
because of the other similarities of the people

around. And so this is cancer ends up being
a really difficult, tough nut to crack. Because

where we see high rates, there are often a
lot of different geographic issues going on

there in cluster sampling doesn't really help
tease

that out.

So to wrap this up, cluster sampling is used
when geography is important. So if there is

something geographically located in a certain
spot and you can't move it, then you kind

of are stuck doing cluster sampling. So briefly,
the map around that areas divided into different

sub areas, right. And those are Not all the
areas are picked, just a few are randomly

picked. And then all of the people in that
particular area are sampled. And of course,

it's biased towards the people living in the
area. If you you know, in the area you pick

with a bunch of affluent people, you'll get
affluent people pick an area with a bunch

of immigrants, he'll get immigrants. And so
a cluster sampling is not perfect, but you're

kind of stuck with it. When there's a situation
with geography, how long it was, remember

it is, when I used to live in Florida, we'd
like to drive up to Georgia because they had

the best pecan clusters. That's like a type
of dessert with pecans and Carmel and stuff.

So when I think of cluster sampling, I think
of those pecan clusters that they're only

really good in Georgia. So that's my way of
remembering that cluster sampling has to do

with geography. Now I'm finally going to talk
about the last two types of sampling that

I'm going to cover in this lecture, convenience
sampling and multistage sampling. They're

both a little quick, so I'm going to just
cover them quickly. First, we're going to

start by talking about convenient sampling.
And we like that name, right? It's convenient.

Convenient sampling can be used under low
risk circumstances, like if the findings of

what you're doing aren't really that important.
Like, for instance, let's say that you wanted

to know what ice cream is the best from the
restaurant next to the hospital, let's say

a new restaurant opens up, and you're gonna
go off your diet, you're gonna go get some

ice cream, but you don't want to waste it
right. So you want to ask people, what's the

best one, you might ask your coworkers, you
might ask, you know, the people at the restaurant,

hey, what's the best ice cream, but the results
are not so reliable, because you might end

up on Yelp and see that other people disagree.
So a convenient sampling is basically using

results or data that are conveniently or readily
obtained. And my master's degree, one of the

things I did was I surveyed people anonymously
who were coming to a health fair, I sat at

a booth, and I gave them the survey, to view
questions in it. That was definitely a convenient

sample, you know, just people showing up for
the health fair. And this can be useful when

there's not a lot of resources allocated to
the study, like, I was a starving master's

student, right, like, I didn't have any money.
So that that was perfect for me convenience

sampling. And also, you know, the questions
I was asking them about were just characteristics

of whether or not they had risk for diabetes.
Well, I'm not a doctor, and I wasn't going

to do anything about it. But it was interesting.
So it wasn't a very high risk survey to fill

up. It and convenience sampling is convenient,
because it uses an already assembled group

for surveys like I was doing at the health
fair. An example might be to ask patients

in the waiting room to fill out a survey or
ask students in a class, you know, sometimes

I do when I'm teaching, I'll do a convenient
sample of whoever sitting there. I'll say,

Hey, is the homework that I signed you this
week too hard?

Well, it's always too hard. I

don't even know why I do the survey. But anyway,
um, sometimes as a teacher, you'll just want

to do a convenient sample just to get the
gauge on where the classes but there are problems

with it, right? You can't just use it for
everything, even though it's nice and convenient.

There's bias in every group, right? So if
I let everybody go on break, and then whoever's

still sitting there, I asked them a thong
works too hard, I might get a totally different

answer than if I waited for everybody come
back. Right. And, you know, just about any

time you just waltz into a room, like when
I went to the health fair, who do you think,

is there a bunch of sick people? No, there's
a bunch of health minded people there. And

so I'm gonna get a bunch of bias, right. And
also, more importantly, when you do convenient

sampling, you often miss important subpopulations.
So remember, stratified sampling, how sometimes

people don't group evenly into the different
strata? Maybe they do kind of in high schools,

but especially when it comes to job classifications,
they usually have fewer bigwigs than they

do. lackeys, right. And if they just have
a few bigwigs, if you do a simple random sample,

you you might miss all of them. So maybe you
try a stratified sample. On the other hand,

if you walk into the break room that is used
by the lackeys and you say, hey, I want to

fill out my, you know, work satisfaction survey.
All of the ones you're going to get are going

to be from the lackeys, you're not going to
get any representation from the upper job

classes because they don't go in that lounge,
so you'd be missing them. So that's the main

problem with convenience sample is the results
can be so severely biased because you're only

asking the small, biased group of people that
probably are all alike in some way. It's not

very representative sample.

Next,

I'm going to talk about multi stage sampling.
So, you know, if you have a kid and the kids

crying somebody like What's up, you say, well,
the kids going through stage as well. That's

exactly what you're doing when you're doing
multi stage sampling, as you're going through

stages. It's basically like mixing and matching,
the different sampling I just talked about,

only you do one stage, and then two stages,
and then three stages, and then four stages,

or maybe even more. And that's how you get
your sample. So if you're imagining why I

got to start with a lot of people, you're
probably right, I just gave an example I made

up of a way that you could do multistage sampling
is you could start one with stage one as a

cluster sample, right? Remember, where you
take out a map, and then you divide into areas?

Well, let's divide into states and take two
census regions of states like about 10 states

from those clumps. Okay, now, we limited it
to that. Now let's go to stage two of our

multistage sampling. Now, from each of those,
we could take a random sample of counties,

right. So we go and look at all the counties
and then take that random sample. Then after

we get those counties, stage three, we could
take a stratified sample of schools from each

county. So some of the counties will be totally
rural, some will be totally urban, but most

will have some mix. So we'll take a look at
a few schools from the urban a few schools

from the rural in stage three from the stratified
will tell you a stratified sample schools

from the simple random sample of counties
from this cluster sample of states. Okay,

now we got our schools, stage four could be
a stratified sample of classrooms. So once

we figured out our urban schools or rural
schools, we could go in there and look at

all the classrooms, freshman, sophomore, junior
senior and take a stratified sample of those.

So it's basically mixing and matching. But
you're right, you got to start with a lot

to begin with, if you're gonna whittle it
down, and a whole bunch of stages, doesn't

have to be four I just gave you for. Now I'm
going to give you a real life example. This

is the National Health and Nutrition Examination
Survey. And Haynes definitely not a Master's

project. This is done by the Centers for Disease
Control and Prevention at the United States,

right. So what I'm kind of hinting towards
is the kinds of places doing multistage sampling

our governments, not only do you have to start
with a whole bunch of people and things and

individuals, states and schools, and what
have you, right, is that it's a lot of work

to do all the sampling, and it better be for
good reason. And the National Health and Nutrition

Examination Survey is a good reason. That's,
that's a survey that's done by the CDC to

try and measure America's Health. Of course,
it's doing inferential statistics, right,

it's taking sample and trying to extrapolate
that information back to the population. And

so it's got to be really careful about how
it does a sampler you can't just waltz in

and do a bunch of convenient sampling. So
this is how it does it, just briefly, they

start by in stage one, sampling counties.
Then from those counties, they sample something

called segments, which is defined in the census,
it's their different areas, from those segments,

those areas, they sample households. And that's
what they mean, like, wherever you live as

a household. Even if you live in a dorm, that's
a household or you live in assisted living,

that's a household. I'm an apartment building
house. So they sample those and once they

knock on your door of your household, they
sample individuals from the house. So they

use four stages of sampling. And that's a
real life example of multi stage sampling.

So in summary, convenience and multi stage
sampling, with respect to convenience sampling,

you want to avoid it unless it's really a
low risk question you're asking about. And

you also want to avoid it unless it's really
the only type of sampling possible under the

circumstances. When you have situations where
you have patients with very rare disease,

probably convenience sampling from your Rare
Disease clinic is reasonable. There, it's

also used when resources are low. And so those
are a few good reasons to try to use convenient

sampling. It's really something that you want
to use only if it's the thing

you're stuck with. It's much better to look
towards these other sampling approaches I

described. And then finally, multistage sampling
is usually used in large governmental studies.

So don't expect to actually design anything
alone with multistage sampling. When that

happens, I showed you those four things for
that survey that the CDC does hundreds of

people work on that even just a sampling tons
of people work to try and set that up. It's

very difficult. But I wanted you to know about
that kind of sampling, because it's important

in healthcare, and it happens a lot. So in
conclusion, we made it through the sampling

lecture didn't wait. I first started by describing
some definitions, you needed to be able to

understand all these different types of sampling.
Then I went into simple random sampling, and

showed you how to do it two different ways
and what it achieves and also its limitations.

We next talked about stratified sampling,
why you do that and how you do that, and the

limitations of that one, too. Then we got
into systematic sampling, which is a little

more flexible, and pretty easy to explain.
Next, we talked about cluster sampling, and

why you might need to pull that tool out of
your sampling toolbox. And then finally, we

covered convenient sampling and multistage
sampling. Already. Well, I hope you better

understand sampling now and can keep all of
these different types of sampling straight

in your mind. Hello, everybody, it's Monica
wahi labarre. College lecture for statistics

are on to Section 1.3. Introduction to experimental
design. And here are your learning objectives.

So at the end of this lecture, you should
be able to first state the steps of conducting

a statistical study, and then select one step
of developing a statistical study and state

the reason for the step, you should be able
to name one common mistake that can introduce

bias into a survey and give an example should
be able to explain what a lurking variable

is, and give an example of that. And you should
be able to define what a completely randomized

experiment

is.

So let's get started. This lecture is in a
cover four basic topics. First, we're going

to look at the steps to conducting a statistical
study, you may think there's a lot of steps

to conducting a study, this is from the point
of view of the statistician. Okay? Then we're

gonna go over basic terms and definitions.
And by now, you're probably used to the fact

that in statistics, certain words are reappropriated.
And they mean something specific in statistics.

So we'll talk about that. Then we'll talk
about bias and what that is and how to avoid

it in when designing your studies. Finally,
we'll talk about randomization in particular

topics you need to think about when thinking
about randomization. So let's get started.

We're going to start with, of course, basic
terms and definitions. And so first, we're

going to review these steps that I keep talking
about to conducting a statistical study. But

there's some vocabulary, vocabulary that comes
up. And so we're going to talk about those

vocabulary terms that come up. And then also,
I'm going to give you a few examples from

healthcare. So here are the steps I keep talking
about. So these are the basic guidelines for

planning a statistical study. So the first
thing you want to do is state your hypothesis.

And you know, I'm in a scientist a while now.
And I can't tell you how many times I get

in a group of us, and people are all curious,
and they start thinking about let's do a study.

And it's only halfway through our conversation
that I suddenly say, Hey, wait a second, we

don't have a hypothesis, what's our apotheosis?
So it's easy, even for scientists to forget

that that's really step one, is you have to
have a hypothesis. And so whatever hypothesis

you pick, the hypothesis is about some individuals,
if I have a hypothesis about hospitals, those

are the individuals I have a hypothesis about
patients. Those are the individuals. But it's

important actually, to nail that down. Because
am I talking about patients in the hospitals?

Or am I talking about the hospitals, so make
sure that you understand after you, you know,

percolate and decide on your hypothesis, who
the actual individuals of interest are? And

that's because you're going to have to marry
measure variables about these individuals.

So step three is to specify all the variables
you're going to need to measure about these

individuals. You know, and of course, they
relate to the

hypothesis.

So it's good thing is that was step one, right?
Step four is to determine whether you want

to use the entire population in your study
or a sample. If you already have a bunch of

data like you have the census data you You
might as well use the entire population. But

typically, if you don't have the data, you're
going to want to sit down and think about

using a sample. And if you do that, while
you're sitting down, you should probably also

choose the sampling method on the basis of
what I talked about in the sampling lecture.

Now that you've figured out your hypothesis,
you got your individuals, you figured out

your variables, and you figured out whether
you're going to do a census or a sample, if

you're going to do a sample what type of sample
Step five is you think about the ethical concerns

before data collection. If you're going to
be asking some sensitive questions, you think

about privacy, if you're going to be doing
some invasive procedures, you think about

how painful that would be, and how hard that
would be on somebody, especially if they're

not even, you know, it's they're just healthy.
And you're just doing an experiment of unhealthy

people just to better understand biology.
So you have to really sit down and think about

these ethical concerns. And they may change
slightly your study design. Finally, after

you get steps one through five, are taken
care of, that's when you actually jump in

and collect the data. And like I was saying,
you know, when I meet with my scientist, friends,

we get all excited about an idea. We're often
talking about Step six, we're like, oh, we

should do a survey, we should this we should
that. And I realized I ended up saying, Hey,

we actually have to go back to step one and
start talking about a hypothesis, because

I suddenly realized, I don't even know what
data to collect, right? If you don't go through

the steps in order, you really aren't doing
it right. Step seven, is after you get the

data, you finally use either descriptive or
inferential statistics to answer your hypothesis.

And that's what statistics is about. It's
here for that. And then finally, after you

use the statistics, you have to write up what
you find, even if you're at a workplace. And

they asked you to do a little survey that
happened once when I was working somewhere.

And they wanted us to do a survey. Their hypothesis
was that they didn't have enough leadership

programs, and they weren't building good leaders
they could promote. And so I was on a team

that did the survey, we didn't, you know,
really publish it, like, everywhere. But we

made an internal report, right. And in that
internal report, we had to do step eight,

which we had to note any concerns about data
collection or analysis, you know, that happened

when we were doing a report. And we also had
to make recommendations for future studies,

or if you wanted to study this in future groups
of employees. So in science, what it usually

ends up being is a peer reviewed literature
report, right? is you do a scientific study,

maybe you get a grant. And then you do all
these steps. And then step eight is where

you actually prepare a journal publication.
And in that, you have to note any concerns

about your data collection or analysis, anything
that might have gone wrong, or not gone exactly

the way you planned, or something you need
to take into account to really properly interpret

what the study found. You also want to make
recommendations for future studies, especially

if you screwed something up, or especially
if you answered a really good question. No

reason to per separate on that question, why
don't we move forward and ask the next one.

Now, these are a lot of steps to remember.
So I'm going to help you try to remember them

in sort of clumps. So let's look at the first
clump, which are steps one through three,

which is data hypothesis, identify the individuals
of interest, and specify the variables to

measure. So let's give an example of that.
So let's say our hypothesis was air pollution

causes asthma, and children who live in urban
settings. You know, that's how we'd stated

or we could say that as a research question,
like does air pollution cause asthma in children

who live in urban settings. And so in that
case, the individuals would be children in

urban settings, and the variables we'd have
to measure our air pollution at least, and

asthma at least. And of course, we'd want
to know more things about these individuals,

these children, we probably measure their
income and where exactly they were living,

and how old they were, and if they're male
or female, and these kinds of things, but

that just kind of helps you think about the
first three steps together. Now let's think

about the second three steps together four,
five, and six, which is determine if you're

going to use a population or sample If it's
sample, pick the sampling method, look at

the ethical concerns and then actually collect
the data. So, when you do that, you can either

quote unquote, collect data, you know, like,
by using existing data by downloading data

from the census, or like Medicare, they have
data sets available that are, are de identified,

so you don't know who exactly is in there.
Or you can collect data yourself, like do

a survey or, you know, get a bunch of patients
that will allow you to measurement. When you

use it, a government data set, often you can
make population measures out of it. And so

you don't really have to go through a lot
of sampling, or ethics, because they've already

provided it for you. And it's confidential.
And that's kind of your data collection. But

most of the time, what you'll see, especially
for studying patients, and treatments, and

cures, and things like that, those are on
a smaller scale. So you end up collecting

data from a sample for those estimates. And
again, you need to choose a sampling approach.

And then you need consent, if legally found
to be human research. So I just want to share

with you in case you didn't know, if you want
to go do research on humans, you're a nursing

student, or your medical students or a dental
student, any any students or or your dentist,

your physician, whatever, a nurse, you can't
just make up a survey, or study design and

go out and do it, you have to get approval
from an ethical board. And that ethical board

will talk to you if what you're doing is considered
li li human research, that you need to get

consent from the patients or the participants
in your study if they're humans. And if you're

collecting data about children, for example,
you have to get the consent of their parents

and the assent of the children. And in the
United States, that way, we have a setup,

it's called an institutional review board
for the protection of human subjects and research

or the short answer is IRB. And so I just
want to make sure that if you ever do design

a study that you know about this IRB thing,
and you realize you have to go through this

ethical board and make sure that they're cool
with it. Before you can move on to the next

step of designing a statistical study. All
right, finally, we're on to the last clump

of steps, which is seven, and eight, right?
So that's using descriptive or inferential

statistics to answer your hypothesis you in
six, you collected the data. Now we're going

to do the statistics. And then step eight
is noting any concerns about your data collection

or analysis and making recommendations for
future studies.

So you can kind of imagine this is where we're
sitting in our offices, and writing up our

research, whether we're writing an internal
report to our bosses, over writing for the

scientific literature to publish for everybody.
So at this point, I just want to remind you

that it matters whether you picked a census
or a sample, for your study design. Because

if you pick the census, you're going to do
a certain kind of analysis. And if you pick

the sample, you're going to do a different
kind of analysis and statistics. So again,

that's all kind of cycles back to your study
design. And what's important here is I want

to talk to you about the two different main
types of studies. Now within these two categories,

you have different subtypes. But these are
the two main types that you can have. The

first is called an experiment. experiment
is where a treatment or intervention is deliberately

assigned to the individuals. So you can kind
of imagine that if you enter a study, and

they assign you to take a drug in the study
that you weren't taking before, that would

be an experiment. But another thing could
happen. I mean, you could do this to individuals,

you could do it to animals, but you could
do it, I keep getting the example of hospitals,

we could choose some hospitals and say, Hey,
you need to try a new policy as the intervention

and and that was assigned by the researcher.
So that makes this an experiment. And the

reason why we have experiments is sometimes
you need them. The purpose is to study the

possible effect of the treatment or the intervention
on the variables measured. And so that's one

option you can do is have an experimental
study where the researcher assigns the individuals

to do certain things in the study. There's
another kind of study The other kind, which

is called observational, and the way you can
think about it is in experiments, the researcher

does something, they intervene, they give
a treatment, right? But an observational,

the researcher doesn't do that the researchers
just observes. So, if you enroll in the study,

and you say, Do I have to take a drug? Am
I supposed to eat something? What am I supposed

to do? And the researcher just says, No, we're
just going to measure you, we're just going

to ask you questions, and we're going to measure
things about you, we're not going to tell

you to do anything different, then you're
in an observational study. So no treatment

or intervention is assigned by the researcher
in an observational study. Now, let's say

you're taking a drug, you know, just because
maybe you have migraines, you're taking a

migraine drug, well, you just keep taking
it, or you can stop taking it, you know, they

don't care, they might ask you about taking
the drug, but they're not going to assign

you to take it. It's an observational study.
I wanted to give you a couple of real life

examples. So Women's Health Initiative up
on the slide was mainly an experiment, okay.

This is was run by the United States government,
but of course, had the cooperation of many,

many universities and, and health care centers,
and most importantly, women. So women in America,

women who were postmenopausal, volunteered
to be in the study. And the study actually

had two separate sections, the experiment
section, and the observational study section.

They really wanted women to qualify for the
experiment, and that the purpose of the experiment

was to study whether hormone replacement therapy,
which is a therapy for symptoms that women

can get if they're postmenopausal, that are
unpleasant. What whether that therapy is good

for women, or bad for women, because they
thought maybe it helps them the post menopause

system symptoms. But they thought maybe it
causes cancer, right? So they know. So what

they had to do was assign, get a bunch of
women who were agreeing, you know that they

would take whatever was assigned to them.
And they had to assign the drug to some of

these women. So that's what made an experiment.
The problem is not all the women qualified

for the study. So they had a separate observational
study, if if the woman did not qualify to

get the experimental drug assigned to her,
then she could be in the observational study.

And because this is these big government studies,
why not, you know, somebody wants to be in

a study, why not study them, just put them
in the observational section.

A very huge, popular long, ongoing study.
That's an observational study, again, run

by Well, this one actually started out of
Harvard. And that's called the nurses Health

Study. Some really smart person figured out
a long time ago, that nurses are, are smart

people, they understand their own health,
they understand other people's health. And

they're good at filling out surveys about
health. So they started studying nurses and

regularly sending them surveys, of course,
they didn't tell the nurses what to do. They

didn't assign the nurses any sort of drug
to take or any diet or intervention or anything.

They just observe the nurses, they send the
nurses a survey, and about the nurses health,

and then the nurse vault fills out that information.
I think it's every two years that they do

that,

they're still doing it.

Also, at this point, I do want to point out
the concept of replication. So just the word

replication, right, regular speaking means
to copy, right? Like, if you ever, you know,

have a new roommate, you might need to replicate
your key. So you have a copy of the key for

the new roommate? Well, part of the whole
science thing is that studies must be done

rigorously enough to be replicated. So those
are little keywords in there. A rigorous study

means one that's done really carefully, like
thinking about sampling very carefully. You

know, like avoiding, for example, non sampling
error not being sloppy, not getting a lot

of under coverage, using a good sampling frame.
You know, I'm just giving you examples that

you might know about. But there's a lot of
things that have to be done in research to

do it properly. It's just like driving or
anything else. You really have to keep your

eye on a lot of different things and you want
to try to do them perfectly. And the main

reason why you want to do that is so if somebody
tries to do this same experiment you did or

roughly the same experiment you did. Because
you can't do exactly the same, right? If I

study this hospital over here, and somebody
wants to study that hospital over there, well,

they're going to get different people in there,
right? But even so if that person decides

that they want to study that hospital over
there, if I did my study rigorously, then

it won't be so hard for that person to replicate
how I did the study. And then we can see if

that person and my study if we get the same
thing, or if there's something slightly off

or what's going on. And so replicating the
results of both observational studies and

experiments, is necessary for science to progress.
So you'll know that a lot of experiments are

done on drugs, before they can be approved
to be given to everybody, because they can't

just do one study, they have to replicate
it, to make sure that the findings are all

sort of coming in about the same and that
we can deduce some information about it, you

really just don't want to rely on one study
for your findings. So I just went over several

steps that we need to follow when we're doing
a statistical study, and we actually have

to follow them in order. And you also have
to determine the type of study you're doing,

you know, is an experiment, or observational
study. And there's a ton of study decisions

you have to make. So you got to keep that
in mind. Now, we're going to talk about avoiding

bias in specifically survey design. Now, you
can do a lot of different kinds of studies.

But let's just talk about surveys, because
that happens a lot in nursing. Nurses interact

with patients a lot, and with the community
with each other. And often they gather information

about those interactions or attitudes or,
or how the healthcare system functions by

using a survey. So surveys can provide a lot
of information and useful information. But

it's important that all aspects of survey
design and administration when you're giving

it, you got to think about minimizing bias
and try you know, try to get a representative

sample trying to get accurate measurements.
And so several considerations should be made.

When you want to think about non response
and also voluntary response, okay, so I talked

a lot about sampling in the previous lecture.
But just because you invite someone to participate

in your study, like maybe you're doing systematic
sampling, and every third patient, you asked,

Would you like to fill out a survey? That
doesn't mean they're going to, right? And

so if that person says no, thank you, even
though there were a sample, that's called

non response. So if I was helping you with
a survey, and you said, Hey, I was getting

a lot of non response, I would look at the
proportion if you approach 200 people, and

80 said, No, you know, that's only a 20% response
rate and an 80% non response rate. if many

people are refusing your survey, the few who
actually completed are likely to have a biased

opinion.

I've noticed this at in in situations where
things are really bad, okay. Like, I remember

going to a subway station and it was flooded,
and it was really in a bad situation. And

there was a man handing out surveys from the
Transportation Authority. And he was like,

please take my survey, please take my survey.
And everybody was waving past him. They didn't

want to grab a survey. While you know me,
I got a bleeding heart for surveys. So I took

his survey, and I filled it out. You know,
I think the transportation authorities not

so bad. Right? I lived in Florida, there's
no transportation there, right? So and here

in Massachusetts, we got a great transportation
system, even if it's flooded or doesn't work

half the time, right. It's way better than
not having one. Well, I'm not the only one

who grabbed a survey a bunch of nice Pollyannas,
like me grabbed a survey. So probably the

Trent Transit Authority thinks that everybody
loves the subway when everybody was waving

past this poor guy because they were so disgusted,
because the station was flooded.

Right? So if so many people are refusing your
survey, a high proportion, the feebly will

actually fill it out are going to be kind
of weird, probably like me. You know, you're

gonna get a bunch of happy people when most
of the people who said no might be sad people.

And so, the reason they may not be completing
your survey has may have to do with how they

feel about your topic. This is not just in
terms of satisfaction. Let's say you want

to talk about how many drinks per night somebody
has. Okay? Do you think a lot of people who

are struggling with alcoholism are gonna want
to fill out that survey? You know, how about

illegal drugs or other illegal activity, people
who are into that they don't always feel so

good about talking about it. And so, you know,
you might get a few people to fill out your

survey, but those are not necessarily the
people who are engaging in the behaviors.

So the fact that we have the freedom to choose
whether or not we want to be in a survey is

great. But from a researcher standpoint, is
you have to be careful. If you get low response

rates, you need to ask yourself who was not
responding? And, you know, am I missing a

good share of opinion there? And then, when
you get people who do respond, you got to

be careful with that two, respondents may
lie on purpose. If you've got a pretty cool

survey, but you suddenly ask a question, that's
too personal. People might just lie. If you

ask, maybe a students you're doing a sin,
you know, maybe satisfaction survey with how

the front desk runs at a dorm or something.
If you, you know, ask a question, have you

ever cheated on a test? You know, my, everybody's
probably gonna say no. Also, if you ask a

question where people don't really know the
answer, offhand, they're not gonna put it.

Like if you ask somebody, you know, when you're,
you know, you asked a kid who's been living

in the house forever, when your parents bought
the house? How much did it cost? I mean, they're

not gonna know. Maybe they'll know, but probably
not. And so you want to be careful when you

design your questions that you're not asking
anything that's so personal, everybody's in

lie about it? Or that you're not asking a
question, then you would have Trump people

try to be accurate, they're probably not even
give you the right answer, because it's just

too hard to think about. Um, respondents also
to, you know, to surveys may lie without meaning

to, like, inadvertently. Again, if you ask
a question about something that happened really

a long time ago, they're not probably going
to get it right. This is called recall bias,

like you can have you can you know how, like,
you can look back at a time in your life,

like, especially if you went through something
really harsh, like if you were a part of a

sports team, and you went to state and it
was really tough that you don't remember the

tough part, right? You sit around singing,
you know, your sports songs, and you say,

Hey, that was awesome. Well, that's recall
bias, right? Because after winning state,

everything looks rosy. But, you know, on the
bus, there really wasn't that easy. So people

tend to have recall bias, it's influenced
by events that have happened since the original

event. So if you're giving people a survey,
and you're saying, Well, before you applied

for nursing school, you know, what did you
think this? Or did you think that, you know,

they might just tell you and think they're
telling you the truth, but they're actually

lying. If you actually managed to go back
in time and ask them, then they tell you something

different. So again, you can kind of screw
up your own data by screwing up your own questions.

So you want to think about how you word your
questions. You can also screw up your questions

by introducing a hidden bias. Something happened
to me recently, where a company sent me a

free app. And they said, try our free app,
and I downloaded it, and it was awful. Okay.

And then about a month later, they sent me
a survey. And these were the questions I said.

When do you use the app? You know, what time
of day? Do you use it? Right? Like how how,

how do you use it? Do you read scientific
literature? Do you read news? And the problem

was, I couldn't really answer any of this.
Because from the day I downloaded it, I never

used it. It was so bad. Right? So question
wording may induce a certain response. They

were asking me how do you use this, but they
didn't give me a choice of I don't. So I had

to say something. I don't even know what I
said. I mean, there was nothing I could say

To be honest, because of that bias. So you
have to be careful that you aren't too rosy

about whatever your topic is, and and assume
everybody loves everything. I mean, you've

got to put out questions like are you even
using the software? Did you have any problems

with the software? Right? I'm just assuming
they're using it and liking it and using it.

You know, like it's supposed to be used is
a big assumption. Order of questions and other

wording may induce a certain response and
you'll see this a lot if you take a public

opinion poll. I used to do a lot of polling
We'd ask questions like, how likely are you

to vote for candidate x? You know, very likely
someone likely? Somewhat unlikely and not

at all likely? And people say, I don't know,
no, no likely. And then you'd say, Well, what

if you knew that candidate x supported this
new proposition? proposition? 69. Right, then

would you be more likely to vote for candidate
x? And so that's why order of questions other

wording and stuff. They're trying to see if
I add this fact that that fact is that going

to make the person like the candidate better.
And so you do have to think about the order

you put the questions. And if you want to
ask about two different subjects, kind of

think about which subject should come first,
because it might color the respondents answering

of the subsequent subject. And also on the
slide, I wanted to point out that the scales

of questions may not accurately measure responses.
Do your feelings always fit on a scale from

one to five? Well, you know, yelps kind of
figured it out. If people's feelings about

restaurants tend to fit on a scale of one
to five, I'd have a lot of trouble filling

that out if they gave me a scale of one to
17. Right. But sometimes people have more

granular feelings about things, maybe they
need a longer scale one to seven. Um, you'll

see a lot of pain scales, where they offer
more than just five choices, because probably

pain can maybe go from one to seven or one
to 10. So think about your scales when you're

creating these questions, because that's your
choice if you're designing the study. Another

point to be made is the influence of the interviewer.
Now, we don't have as much interviewing going

on these days, because we have the internet
where we can do anonymous surveys, and people

just fill them out self report, we have Robo
phones that you can call robo call. And using

an automated voice, that's obviously not a
person, you can get survey data. But there's

always situations where you actually have
to interview people, especially if somebody

is really sick in bed, and you have to show
up there, you have to talk to them. And so

even on the phone, you have to interview people,
and they can hear your voice, right. So you

got to think about when you're pairing up
whoever's being interviewed with whoever's

interviewing, um, I've found that it's best
to have the interviewer come from the same

population as the research participant, in
general, the only time that can be a problem

is a thirst from the same community, and there's
a privacy issue. But it can be very helpful,

for the most part, not always, to have your
interviewers be actually from the population

that you would be studying, you know, from
the individuals that you would be studying.

So for instance, if you need to interview
a bunch of young African American, you know,

like some African American teenage men, like
I recently saw a study on how health care

in the United States really isn't suited for
them. And it needs to improve and needs to

better cater to this population. Well, let's
say you wanted to better understand that,

the best thing would be is to hire a young
African American male and train him on how

to be good interviewer and do be good data
collector, because you probably get the best

data that way.

On the other hand, let's think of different
ways that that could go, you could take a

person who was older, who is maybe of a different
race, and maybe that would change how this

young African American male would respond
to this interviewer. I mean, the interviewer

could be like, in many ways, like the respondent,
but the respondents perception might change,

then how they answer all verbal and nonverbal
influences matter, you know, clothing, the

setting that the person's being interviewed
in. And so I'm not saying there's really a

solution to all this. I'm just saying, make
some good decisions. Like I remember working

on a data set where there were some questions
that had been asked about some older men about

their sexual function. And I, it looks the
data look funny to me in the statistician

who was there during data collection told
me that they had chosen young, female nursing

students to interview these elders. Men about
their sexual habits. And I just said, you

know, that might be subject to interviewer
influence. And then you of course have to

worry about vague wording. Just because it
looks clear to you doesn't mean it looks clear

to everyone. There are simple ways of avoiding
vague terms in the survey, when you can just

put a number on it. So instead of asking a
person, if they've waited a long time in the

waiting room, you can say, more than 10 minutes.
You can say exactly like within the last month,

have you done certain a certain activity or
within the next year? Do you expect to change

schools or whatever. And so try to wherever
you can use numbers or something very specific,

you know, instead of go to the clinic, go
to the public health clinic at this particular

corner, or whatever. And then you're going
to get some pretty accurate information.

But

sometimes you're stuck using vague terms,
because you're studying vague terms, right?

I was doing a study of controllable lifestyle
attitudes towards controllable lifestyle in

medical students. So we asked this question,
how important is having a controllable lifestyle

to you in your future career? Well, what does
that mean? That's pretty vague. So what we

did is we use this grounding this anchoring
language,

we added the sentence, a controllable lifestyle
is defined as one that allows the physician

to control the number of hours devoted to
practicing his or her specialty. So even though

we're talking about something kind of wofully,
and watery, loosey goosey like control of

a lifestyle, who knows what that means? And
that's not to say that that sentence could

be interpreted differently by people it certainly
is. But if you're stuck with vague wording,

try to put some grounding language in it.
So everybody's at least sort of led in the

same direction with their thought before they
answer the question. Now, I want to also point

out, you probably have noticed, there's all
these issues, you have to think about when

doing surveys, there's this other issue called
the lurking variable, well, you know, lurk

means to sneak around behind the scenes, right?
Behind the scenes, a lurking variable is a

variable that's associated with a condition,
but it may not actually cause it. I remember

when I was studying epidemiology, they talked
about how a lot of people with motorcycle

accidents, you unfortunately got in motorcycle
accidents that they had tattoos. So therefore,

they said, Everybody shouldn't get a tattoo,
you might get it in a motorcycle accident?

Well, that's a great example of a lurking
variable. Yeah, a lot of people who do get

into motorcycle accidents, have tattoos, but
that the tattoos don't cause that. Um, we

also know that having more education increases
income, but people have the same education

level do not all make the same income, there's
this thing, you know, called, it's sexism.

And it's called racism. So it matters whether
you're a woman or a man, it matters, the color

of your skin. If the you know, if you've got
a darker skin, doesn't matter, that you have

the same education as somebody with lighter
skin, you're still gonna make less money.

And so you have these lurking variables behind
the scenes. So when people are looking at

Well, why are people you know, making less
income, because they're less educated, whatever?

Well, you got to look for also the lurking
variables. So current studies show that why

women and African Americans make less money
on the whole, it's not explained by fewer

of them working or fewer of them getting degrees.
It's really these lurking variables. And so

you got to think critically. And I guess what
I would say is, whenever you do a survey,

if you're studying something that has a lot
of lurking variables associated with it, make

sure you measure those variables. Like early
studies where they were looking to see if

drinking a lot of alcohol causes lung cancer.
Some of them forgot to really study how much

these people would smoke. Because we know
smoking causes lung cancer. And we know if

you're hanging out in a place with a lot of
drinking and they allow smoking, you'll see

a lot of people smoking too. They seem to
go hand in hand. So you don't want to miss

measuring variables that you think might be
lurking variables. It's no problem to measure

them and not use them later, but just make
sure they're included. So, as a final note

on bias, I just want to point out that survey
results are so important. for healthcare,

and for the progression of science, that you
really owe it to even a simplest survey, to

think about all of these things, these possible
things that could go wrong, just with the

wording of questions or with how you're approaching
things, and just really consider how you can

improve it. It's really important to pay attention
to avoiding bias when you're designing and

conducting your survey. So think about all
these things at the design phase. Finally,

I'll get into the last section of this lecture,
which is about randomization, which I think

a lot of us have heard about. So I'm going
to explain the steps to a completely randomized

experiment. And after I go through all that,
I'm going to also talk about the concept of

a placebo and the placebo effect. Then we're
going to briefly touch on blocked randomization,

and also define for you what is meant by blinding.
So why ever randomize, right? So what randomizing

is, is when you take a bunch of respondents
or participants in your study, and you randomly

choose what group they go in. And if you remember,
like I was talking about experiment versus

observational study, we can't do that in observational
study. This is definitely an experiment because

you're telling them what group to go,

right. So randomization is used to assign
individuals to treatment groups. And when

you do that, when you randomly assign them,
not only you're assigning them, but you're

randomly assigning them, you're not picking,
you know, you're using like dice or some sort

of random method, and helps prevent bias and
selecting members for each group. It distributes

the lurking variables evenly, even if you
don't know about the lurking variables, even

if you aren't measuring them. By using this
randomization method, they get equally allocated

in each group. So just to remind you, how
you actually do that is, first I remember

the steps to that statistical study, you have
to follow those. And after you get to the

point where you have ethical approval, that's
when you start doing the data collection step.

And that's where you start recruiting sample
or, you know, hanging up signs and saying,

Be in my study, and people come in, and you
see if they qualify, and if they qualify,

you've got this group of sample, right. And
what you do with those people is you say thank

you for being in my study. And you measure
the confounders, which is another word for

lurking variables. You also measure the outcome,
whatever you're trying to study, if you're

doing a randomized experiment, I know I've
been involved in a lot of these where they're

studying drugs for lowering blood pressure.
So they'll often have maybe two groups or

three groups, where they're randomizing people
into, but they don't do that first, the first

thing to do is get everybody in there and
measure their blood pressure, right? The outcome,

you know, because they want to know that before,
they are going to take a picture of that before.

And they also measure confounders, like smoking,
remember, smoking is not good for your blood

pressure, you know, other things are not good
for your blood pressure, like not exercising,

well measure all of those things. Okay, now,
here's where we get into things. That's when

the whole randomization happens. So I showed
this picture of a dye, but we usually use

a computer for it. So we got all these people
together. And now you know, randomly, we put

them in different groups. And in this example,
on the slide, we're just going to pretend

that there's two groups. And in fact, we can't
really study blood pressure on the slide.

Because we're going to give one group treatment
and the other group placebo, which is an inactive

treatment, it's fake, it doesn't work. Of
course, the treatment and the placebo are

going to look the same to the people taking
it or, you know, we're going to fool them.

They don't, they won't know. But the reason
why in real life, you can't do that with a

blood pressure study

today

is we know that high blood pressure is really
bad for you. So it's really unethical to give

someone a placebo, you got to give them some
sort of drug to lower the blood pressure.

So usually when we do studies like this on
blood pressure, now, new blood pressure drugs,

Group A is treatment in Group B is old treatment,
like they usually take a new treatment and

give it to group by an old treatment to Group
B, see if they can find just a better treatment.

But if we were talking about something like
all timers, especially late stage old timers,

there's no treatment. Okay? And so what go
what's on the side here, Group A, that gets

treatment and Group B, which gets this Sham
pill, this placebo, that would be ethical

then, but let's just cross our fingers that
someday that's not ethical anymore and that

we do get a treatment right.

Okay. So

after you put them in the two groups with
sort of missing from the slide is time passes,

people in Group A take whatever they're supposed
to take their treatment. And in this example,

on the slide, people in Group B, take the
fake treatment, the placebo, and neither of

them, you know, usually knows what's happening.
But it takes a while, right. And in the olden

days before we knew high blood pressure was
bad. These were the study designs. And this

is what ended up happening is that you would
see, at the beginning where they measured

the confounders and the outcome, everybody
had high blood pressure, they all look the

same. But after treatment, Group A would go
down, whereas group and Group B would go down

a little bit from CBOE effect, which I'll
explain in the next slide. But that's how

we learned that you can make blood pressure
go down with these different pills. Finally,

after that time passed, it could be six weeks,
it could be years, however long that took

after that passed, when it was over, we'd
measure again, the confounders because they

could have changed. And the outcome, which
in my example, was blood pressure, or, you

know how serious some of these Alzheimer's
disease would be, if we were doing that. So

I promised you on the last slide that I talked
to you about more about what a placebo is,

and the placebo effect, found this great picture
of old placebos from the National Institutes

of Health. So a placebo is this fake drug
that's given and it's actually kind of hard

to make placebos. Just imagine a drug you
may need to take me even excetera and or something

like that. Imagine we had to study etc. And
we'd have to make a fake excedrin that tasted

like it and look like it. Because then Otherwise,
the people who are randomized to the placebo

group would be able to totally tell that they
were in the placebo group, and that's not

good to do. So, what the reason why you need
a placebo is there's this thing called the

placebo effect. And that occurs when there
is no treatment, but the participant assumed

she is receiving treatment and responds favorably.
Now, sometimes I talk about one of my favorite

epidemiologists, comedians, Ben Goldacre,
he reported in one of us, I think one of his

TED talks about a study where they everybody
they enrolled, um, they didn't have a disease,

right, I guess they had a mild disease. And
they told everybody, either they were going

to give them nothing, or they were going to
give them a pill, that's a placebo, it doesn't

do anything. Or they're going to give them
an injection. That's a placebo injection,

it doesn't do anything. And what they found
is of the three groups, the people who got

the injection did the best. And the people,
you know, the fake injection, people got the

fake pill, the placebo pill, that is second
best that people didn't get anything didn't,

the worst. And that his point is, that's what
the placebo effect is, for some reason, when

we're getting injected. Even with just sailing,
we think we're getting some sort of drug and

it psychologically, or however, affects our
bodies. The same thing when we're taking a

pill. I don't know if you've ever seen kids,
you know, saying, Oh, I need medicine 90 minutes.

And then then the parent gives them an m&m,
right, they think it's a pill, they're happy

with it. But actually, the placebo effect
can cause real effects on your health, it

can make you feel better just because you
think you're taking a drug. And so that's

why it's super important to include a placebo
group, if you don't have a comparison group,

like I described with blood blood pressure
in all your studies, because if you just have

one group where they're taking it, they'll
all say it's good. They would say it's good

if it was water, right. So the placebo is
given to what's called a control group, and

they receive the placebo. Now, if you're studying
like acupuncture, you can't really give up

placebo acupuncture. So what they'll do is
they'll sort of hang, hang up a little curtain

and kind of tap you and you don't know whether
you're getting real or it's called sham acupuncture.

Other things have to happen like that when
you're doing these studying these interventions

that aren't pills. Those are called attention
controls, right? Where we have like a sham

acupuncture. So in any case, you've got to
think about this because you need a controller

comparison group. That's fair. Whenever you're
testing in an experiment in a randomized experiment,

a new thing

promised you I'd talk a little bit about blocked
randomization, I won't get much into it. But

sometimes when you go to randomize, right,
you know, you get this whole group of people,

they're all about the same, but you're gonna
split them into a group A and Group B, one's

gonna get maybe a drug and the others maybe
gonna get the placebo. Sometimes you get worried

that the groups are going to be unbalanced
with respect to a particular lurking variable.

In blood pressure, we'd always care about
smoking, we want the equal amount of smokers

in each group. You know, a lot of times we
we care about gender, we want equal amounts

of men and women in each group. So if you're
worried about that, with randomization, you

can't just do it one at a time, because you
might just randomly put too many men in one

group. So what you have to do is block randomization.
So see, I drew all these blocks on the on

the screen, and you'll see that there's nobody
in them, they're just blank, I just put xxx.

So this is before you do your study, you have
these blank blocks. And what you do is as

you enroll those people remember you have
to measure them and make sure that they qualify

for the study, as you get them in, you can
just write them in the blocks, right. So here,

I just put their fake initials, you know,
so let's say that XYZ came in first, that's

a woman, and then maybe NSW came in, and that's
another woman, you just keep putting the women

there. And then when the men come in, you
put them in, and you fill up the blocks, then

here's a trick, you actually randomize the
entire blocks, right? So block one and block

three ended up in Group A, and but magic,
you got to equal men and women there. And

then Group B equal men and women. And so that's
how you do with blocks. So but you know, there's

some limitation to this, like, if you get
multiple races in your study, maybe, you know,

four or five racial groups. If you make a
five block, you've got to fill up the whole

block before you randomize it. And, you know,
sometimes you're you're in an area where certain

racial groups are rare. And you might have
trouble filling up your blocks. So there's

some limitations of this too.

Now,

I had mentioned the situation where you really
don't want if you're going to do an experiment,

right, not an observational study, experiment.
And you're going to randomize people either

to a drug or some sort of intervention versus
placebo, or a drug versus another drug, an

old drug, you really don't want them to know
what group they're in. I mean, because you

have to be ethical. before they enter the
study, you have to tell them, you're gonna

put them in one or two group, one of two groups,
but you got to tell them, you're not going

to know what group you're in wallets going
on. So blinding is where the, where any person

is deliberately not told of the treatment
assignment. So he or she is not biased in

reporting study information. And it actually
doesn't have to just be the participant in

the study, it can be researched, like, the
most common one is a participant is blinded

to treatment or placebo. But I've been in
studies or I've been worked on studies of

like Alzheimers disease, right? Well, they'll
they want to take the patients are the participants

in the study might have Alzheimer's disease,
and look at their image, the MRI of their

head. And often, they'll have also a neurologist
interview them, they'll also see a neuro psychologist.

And they often want those three different
groups, they imaging group, the neuro psychology

group and the neurology group, not to know
about each other's opinion of this particular

patient. So they'll blind them to each other's
opinion. So blinding AR is much more complicated

than just blinding the participant to whether
or not they're in placebo, or they're in drug

group. But double blind is a really important
concept. And that means that both the participant

and the study staff do not know the treatment
assignment. So everybody who's operating with

the patient doesn't know it. So you're probably
thinking that's really pretty serious, right?

Like, what if that person gets sick, and goes
to the emergency room, and they're taking

an experimental drug or they could be taking
placebo? Who knows what they're taking? Well,

in that case, what happens is there's an unblinding
procedure, there just has to be as part of

ethics. It's already set up in the study.
If somebody goes to the emergency room, there's

a person that can be called to unblind. The
pate, the participant who's now a patient,

and and once they're unblind, they learn what
they were taking. Even if they were taking

placebo, the whole thing's over. Right? Even
the study staff work. It's just a fact of

life. It has to happen sometime. But for the
most part, what we tried to do is keep things

steady. double blind because it makes things
the least biased in the most fair. So 10,

the session on randomization, the purpose
of randomization, why we go through all this

when we're testing treatments, especially,
is that it's used to reduce bias. And especially

if you have a particular variable you're concerned
about like gender, like we were talking about

race, or smoking, smoking status, you can
use a block randomization to even out each

group. And then blinding further prevents
bias, right? Because people don't know what

they're taking in the study staff don't know
what they're giving them. And the reason why

you have to really think about blinding is
the placebo effect is necessary to take into

account, you're always going to get the placebo
effect every time you give somebody something.

So you've got to account for that in your
study design. So in conclusion, I went over

the steps to conducting a statistical study
in order and kind of give you tips on how

to remember that we looked at some basic terms
and definitions. And we talked about how to

avoid bias in survey design, because there's
a lot of different considerations. And finally,

we talked more in depth about specifically
about randomization in experiments. All right.

Now, you know, a lot, maybe too much. I hope
you enjoyed my lecture.

Hi, Whoa, it's me again, Monica wahi, your
statistics lecturer from labarre College.

Now we're going to go go back and cover what
I didn't cover in the last lecture about chapter

2.1, which are frequency histograms and distributions.
So here are your learning objectives for this

lecture. So at the end of this lecture, you
should be able to state the steps for drawing

a frequency histogram, you should also be
able to name two types of distributions and

explain how they look, you should be able
to define what an outlier is, and say one

reason why you would make a frequency histogram.
Finally, you should be able to define what

a relative frequency is and what a cumulative
frequency is. Okay, so let's get started.

First, we're going to review frequency histograms
and relative frequency histogram. So you'll

figure out what I'm talking about there. Then
we're going to go over five common distributions

in statistics, so you know what that's all
about. And then I'm going to talk about outliers.

Now, you'll notice I have a lot of pictures
in this presentation of skylines. And the

reason why is they remind me of histograms.
So let's talk about what is a frequency histogram.

So a frequency histogram is important in statistics,
because, as you'll see, you need to make one

in order to see what the distribution is.
So I'm going to go first explain what one

is, like, show you what one looks like. And
then I'll explain how to make one. And then

I'll explain the relative frequency histogram.
And then we'll move on to looking at why do

we need that for distributions. So here's
another skyline because it looks like a histogram

to me. So what is a frequency histogram? Well,
it's actually a specific type of bar chart.

And it's made from data in a frequency table.
So you might see a frequency histogram and

go, well, that looks like a boring old bar
graph. Well, it's not just any old bar graph,

it's got specific properties that I'm going
to talk to you about in this lecture. Okay.

Both frequency histograms and relative frequency
histograms are bar charts with their special

bar charts that have to be done a certain
way. And why? Because if they're done that

way, in their histograms, they will reveal
the distribution of the data, which I'll explain

later. So here is a frequency table, we had
this before. This was of those fake patient

transport miles, right. So you'll notice here
were the class limits, and then we put in

the frequency and we even threw in this relative
frequency. Okay, so this is the frequency

table I'm going to use as a demonstration
for how you make a frequency histogram, you

first need a frequency table. Okay, now, here's
the histogram version of what's in that frequency

table. So I'm going to annotate this one image
to explain the order in which you draw it

basically by hand. So the first thing you
do is draw this vertical line for the y axis,

okay, you just draw a line.

Next, you write

words next to the line, and you always start
with frequency of, and then whatever In our

example, it was patience, okay. And I'm telling
you, you need to do it in this order, or you'll

get confused. So you start with that first
line, and then you write this frequency. Okay.

Next, you draw the whole horizontal line for
the x axis,

okay.

And then after that you write the classes
below. Remember, like the lowest class is

one to eight, that's a lower class and an
upper class limit of the lowest class, like

you literally write those labels in. And why
do I, why am I so freaking out about this

order is because I totally get confused if
I do not do this y axis first. Because then

all there's all these numbers. And it's totally
confusing. So just try to do it in this order.

Okay. Now, number six, I had to flip the slide
here. Okay, at step six, use drawn like the

basic background, you've got the x and y axis
and those labels. So now you have to start

drawing in the bars. So for your first bar,
you look at the first class, and you find

the frequency on the table, which I think
it was 14 or something. And so you look for

it on the y axis, and you want to label the
y axis so that the maximum one is is incorporated

in it, like you see our maximum is above 20.
So we wouldn't want to end our Y axis at 20,

or 15, or something, you have to make it bigger,
so you can put everybody on there. But our

first one was what at 14, so we draw this
horizontal line around the 14, right there,

that that horizontal line, because we're gonna
make that first bar,

then

you draw the two vertical lines down, and
you position it over where you labeled the

class. And that makes the bar and then you,
you actually color in the bars, like and you

repeat this for each class, right? So you
go, that's why I labeled the classes first

on the x axis just to make sure everything
is even. And then I go through and I make

all the bars. And again, this is why you need
to prepare your frequency table first. So

you know how to graph it, you know what to
put on this graph? Okay, this is the relative

frequency histogram, you already understand
what relative frequency is, right? It's that

proportion, the proportion of your sample
that's in each class. And so the change, if

you're going to do a relative frequency histogram,
you basically go through the same steps, it's

just you're changing what's on the y axis,
you change what you label it, okay? But the

x axis stays the same. And even though you're,
you're charting the relative frequencies,

like, you'll be like, Okay, this is a totally
different number, what you'll see is the pattern

ends up being the same. So it takes on the
similar pattern, which is the pattern is actually

what we're going after, that's the thing I'm
going to talk about with a disparate distribution.

And so I tend to prefer since the pattern
is going to come out the same, I tend to prefer

using a relative frequency histogram, versus
a frequency histogram. Because if I have two

different groups, like let's say, there were
two hospitals, and I gathered two sets of

data, and I wanted to compare the models transported,
then I could use this relative frequency histogram,

and not only with the patterns be evident,
but I could compare them fairly, like whatever's

35, you know, point three, five or 35%. In
this, even if the other hospital maybe had

tons more transports, I could see it as like
35%. And I could really compare the percent,

right. So that's why I lean towards relative
frequency histogram. But ultimately, you're

going to get the same pattern on your histogram,
whether you use frequency or relative frequency.

So again, another picture of a skyline. So
you can see why I think of skylines because

they look like histograms, right? So after
making a frequency table, what you do with

quantitative data, right? Because you're trying
to organize it, it's also important to then

make a frequency histogram and or relative
frequency histogram, and why it's because

it reveals a distribution. And now, that's
what we're going to talk about. We're going

to talk about distributions. So first, I'm
going to define what I'm talking about with

the distribution. And now you're gonna see
a lot of other kinds of pictures like this

on the right, see that that shape? That's
one of our distributions, okay. And so that's

a little prequel to what I'm going to say.
So first, we're going to talk about what these

distributions are. Then I'm going to describe
what an outlier is, and, and how you can detect

them by using histograms. Finally, I'm going
to wrap it up by explaining what cumulative

frequency is and when an old jive is.

Okay, so what is this distribution thing I
keep talking about? Well, it's actually just

a shape. It's the shape that is made if you
draw a line along the edges of the histograms

bars, so On the left, you see I drew the scribbly
shape. But you'll notice you can do it with

a stem and leaf too. This is not the same
data graphed on the right in the stem and

leaf. I'm just using, you know, recycling
the old picture that I used before. But you

see, you can do the same drawing that squiggly
line, you know. And that's actually the distribution.

I mean, they don't all look exactly like that.
But that's what you do is you draw this line

thing. I know, it's kind of odd that that's
what a distribution is, is just a shape. But

there's actually five of them that we use
a lot. There's way more than five, actually,

in statistics, but you have to get into kind
of higher level statistics to care about those,

we're only going to concentrate on these five.
Okay. So the first one is called normal distribution.

And it's called that everywhere, except I
noticed the book call that mound shaped symmetrical

distribution, but I'm going to call it a normal
distribution. And there's nothing really normal

about it, it's just named that for some reason.
And then there's a uniform distribution, skewed

left distribution, skewed right distribution,
and by modal distribution, so those are the

five we're going to cover. So let's start
here with the normal distribution. So as you

can see, on the right, somebody made a histogram.
And then they do that squiggly line. Well,

actually, it was me who made this histogram
and drew the squiggly line. And notice the

squiggly line, what it looks like, it kind
of looks like what the book called it, it's

mound shaped and symmetrical. But that's the
shape of the normal distribution, it looks

like that it's got kind of hokey things on
the side, and, and a mound in the middle.

And if that's what your histogram ends up
looking like, where it's kind of like a little

mountain like that, then you've got a normal
distribution. Okay, let's look at a different

histogram. Okay? In this histogram, you'll
notice that like, each of the bars, each of

the frequencies is almost like the same, right?
It's either five or six. And it doesn't matter

what class we're talking about. When it's
like that, the little line you draw across,

it's not squiggly at all, it's straight. I
don't see this very often in healthcare data.

But it does happen in other kinds of data
more frequently. And this is called the uniform

distribution, which makes sense, it's almost
all of these bars are a uniform height. So

that's what a uniform distribution is. Okay,
now, this is one kind of like the one we were

looking at before, where it looks kind of
like a slide like at a playground, where,

you know, like, you climb up the right side,
and then you slide down to the left side.

Okay? And that whenever it's like that, where
it's low on one side and high on the other,

it's called skewed. The problem is, which
way is it skewed? Right? And how I remember

which way to say it's skewed? Is it skewed,
where it's light or short? So here, I would

say it's light on the left. So it's skewed
left, right? Because on the left side, it's

really the bars are all short. And then you
can just imagine what's going to come next

here? Well, look at this, this is skewed,
right, because it's light on the right. It's

short on the right. So it's skewed, right.
So technically, I mean, both of them are just

skewed distributions. I like I just like to
explain them separately. Because sometimes

people don't know which way to say is left
to right. And this is how I remember light

on the left, light on the right. Finally,
we have bi modal. Now, the word mode in some

areas of statistics, and then engineering
and stuff often means like a high point. And

by modal means two high points. So as you
can see, it looks like a camel with two humps.

And it's a little hard sometimes to tell by
modal from normal. Because if you remember

normal, like let's say you have a normal distribution,
but you just have one little

one little bar kind of in the middle, you're
like, is this bi modal, or is this normal?

How I tell coach people to see if it's bi
modal is if there's a really big space between

the two humps that's not so apparent on this
image here. But you'll see class three and

class four, they're both short. If only one
of them was short, I might I might have called

it a normal distribution. But I've really
seen by modal distributions when it comes

to like lab data, because my best friend is
a pathologist, and he'll show me you know,

with situations where people have like really
super high platelet counts, and then like

no platelets practically and there's nothing
in the middle. And that's where you'll see

a bi modal distribution. Now we're gonna talk
about outliers. And outliers are data values

that are, quote very different from other
measurements in the data. What's very different,

right? Like it's an opinion. But people in
statistics come up with different formulas

to try and figure out if something is very
different from the other measurements. And

we'll talk about that actually, later in later
chapters in the class, not so much for identifying

outliers, but just to just to better understand
our distributions. But just as a quick and

dirty representation of what would be an obvious
outlier lit, like nobody would disagree on

is this histogram here. So you'll notice I
just threw down nine classes, I made up this

data. But you'll see a class two and class
three, there's just like nothing, and there's

nothing in class eight. But when you get,
and then suddenly, there's something in class

one and something in class nine. And when
you have these big gaps, this is kind of like

that platelets, like I was telling you about
only this maybe would be you know, you would

say this is tri modal, like there's three
modes, but there's not really three modes,

right? There's a wacky low one and a wacky
high one, and everything else is in the middle.

So because that one in class one, and that
one, and class nine, they're so far away from

what's in the middle, like just about every
statistician would agree, these are both outliers.

But you can just imagine how much we argue
about what actually is an outlier. It's especially

hard when you're getting data on weight of
people. Some people really do weigh 400 500,

maybe even 600 pounds, you don't know if they're
really outliers, or data mistakes, or what

to do with them. They're real people. And
maybe they have really high weights. And unfortunately,

some of them have really low weights too.
So the one of the main points of doing the

histogram is not only to look for these distributions,
but also to see if you've got any super obvious

outliers that you're just gonna have to think
about before you proceed with your analysis.

Now, I'm going to talk to you about what cumulative
frequency means, you know, the word accumulate

means to just like keep accumulating things
like if you have a gutter on your house, it

will accumulate leaves, like old leaves will
sit there and new leaves will keep coming

and the old ones will still be there, until
it like totally clogs your gutter, and you

have to clean it. So that's what cumulative
frequency is, is where it accumulates all

the frequencies. So you see on the slide,
you know, in the first class, when they ate,

we had a frequency of 14. So your cumulative
frequency, those are like the leaves at the

first beginning of the season, that's all
you got is 14. But when you add on the next

class 21. Now you add to the cumulative frequency,
it accumulates, you add that 21 to the 14,

and now you've got 35. And if you can extrapolate
as you walk up all these classes, eventually

you get to the total, right. And so yeah,
so that's what you got. And the first class

is always the same as the frequency and each
cumulative frequency is equal to or higher

than the last one.

I'll have to say in healthcare, we don't really
use cumulative frequency a whole lot, you'll

see it but we are really into relative frequency,
I'll just tell you that. But some groups are

into cumulative frequency and those who are,
they like to plot it in a plot called an Ojai.

And again, I'll be honest, and healthcare,
I've never seen an old giant that was just

in the scientific literature, which is why
you'll see this is about NFL teams salaries,

because I think they use it a lot more in
economics. But at any rate, what you'll see

is that the classes are along the x axis,
you know, you're used to that, because that's

what we do in a frequency histogram. But along
the y axis, you see these numbers called cumulative

frequency. And you just graph it, right, but
one of the things you'll just notice is that

it's going to go up, like each one is going
to either, unless you have a class with zero

in it, it's going to stay the same for that
one. But otherwise, it's just going to keep

going up. So you'll always see some sort of
shape like this, where it's always going up

and it hits the top. At the end, it hits the
total cumulative frequency at the end. So,

just to review, there are five main types
of distributions used in statistics. And I

emphasize mean, there's other ones, but these
are the ones we're going to look at. And so

that's why we were doing our histograms and
our seven leaf displays is we were looking

for these distributions. And also we were
looking for outliers. And then finally, I

just quickly did a shout out for your Oh,
jive here and your cumulative frequency. So

you know what, what's up with that. So in
conclusion, the purpose of the histogram is

to reveal the distribution and also the stem
and leaf displays reveal the distribution.

And you look then, for outliers. You'll probably
wondering, Well, why do we do all this work

to, to reveal the distribution, we'll you'll
find in later chapters and matters, what kind

of distribution you have, what kind of statistics
you can do insert, in a way, you know, like

I went kind of, on and on about the normal
distribution. Well, we all really like that

in statistics, we're all really partial to
that, because it allows you to do a whole

bunch of different statistics, you know, pretty
easily if you get a normal distribution. However,

what's often happens is in healthcare, because
I've done it, is you get a skewed distribution

left skewed right skewed, and then you have
to make some decisions, that makes it a little

harder. Also, I've had to buy moral distribution
before I'm remembering that one day, that

was kind of an issue, and then I had to figure
that one out. So that's roughly why we have

to go through this chapter and figure out
how to do these distributions. And then later,

I'll explain to you what you do with that
knowledge. Hello, there, it's Monica wahi

labarre College statistics lecturer. We're
going to circle back now to chapter 2.2. And

talk about these other graphs, I'm doing things
a little out of order, because it makes sense

to me. I hope it makes sense to you too. Well,
for this lecture, we're going to have these

learning objectives. So when you're done with
this lecture, you should be able to describe

a case in which a time series graph would
be appropriate, you should be able to explain

the difference between what would be graphed
on a bar graph versus a time series graph,

you should be able to describe the type of
data graphed in a pie chart. And you should

also be able to list two considerations to
make when choosing what type of chart to develop.

Alright, so let's get started here. What I'm
going to be doing it in this lecture is, first

I'm going to explain what a time series graph
is. Then I'm going to talk about a bar graph.

And of course, I'm going to show you roughly
how to make these, I'm gonna explain a pie

chart and how to make that. And then I'm going
to go over a review of all the graphs I've

talked about for chapter two. And just summarize
when to use what type of graph. So let's start

with the time series graph. And actually,
the word time is the key.

The time

we're going to talk about this time series
graph and what our time series data, right.

As you can see, by this little example, time
is across the x axis. And that's kind of a

hint for where we're going. Okay, so then
I'll show you roughly how to plot one. And

I'll explain why we have these time series
graphs, like how you interpret them and why

you even make them. So, of course, I'm an
epidemiologist. So what am i into m&m mortality,

morbidity. So here's a nice time series graph,
wonderful graph of the percentage of visits

for influenza like illness reported by the
US outpatient influenza like illness surveillance

network, by surveillance week, and this is
October 1 2006, through May 1 2010. And you're

like, oh, time? Yeah, that's the deal. time
series data are made of measurements for the

same variable, for the same individual taken
in intervals over a period of time. Only.

In this case, in the example here, the individual
is not a person, right? Because remember,

individuals are just what you measure what
you're measuring variables about. Here, the

individuals are actually weeks, right? Because
every week, they're making a measurement.

So like I said, time series data are made
of measurements for the same variable, which

is what percentage of visits for influenza
like illness. So every week they went to I

don't know who is in like, what clinics are
in this outpatient influenza like illness

surveillance network, but let's just pretend
there's like 10 clinics in there. So each

week, these clinics have to go in and say,
Yeah, I had, for example, 100 visits this

week, and 10 of them were for influenza, like
illness. So then that would be 10%. That week,

for that clinic. Well, they got all the clinics
together, and they found out what the percents

were. And you can see on the y axis, right,
there's the percentage, and then you see on

the x axis all the weeks in the year. So um,
so you've seen these before, right? You especially

see it with stock market, right? You go on
Yahoo, and look at your favorite stock, right?

You know, we're also rich, we own so much
stock, and so you track your favorite stock

that way. Personally, I'm spend more time
looking at mortality and morbidity, things

like influenza, but hey, there after I get
some money, I'll be looking at stock market

prices. So when we see these time series data
graphed in these time series graphs It's often

about things like influenza rates. Other rates,
you'll see life expectancy, rates of heart

attack. And that's usually what we see, because
we're trying to affect those rates. And we're

trying to see if they're going up or down.
So I'm going to just roughly go through how

you make one, if you ever wanted to make one,
the first thing you need is a table, kind

of like the one on the right, I just made
up these data, they don't mean anything. But

roughly what you need is a column that says,
in this case, I put year, the influenza people

they put a week, but you have to put like
regular time increments in the first column.

And then you have to put that variable measured
at that time in the next column. So let's

say it's today, and you're like, Oh, I want
to measure how many times I went to the gym

each week, you know, weekly over the last
few months? Well, you're gonna have to reconstruct

that data, right? Like maybe from your memory
or your calendar. So normally, when you're

going to go do time series stuff, you start
and you collect the data as you go along.

And then it's nice and accurate. Okay, so
let's say you did that, and you managed to

get some time series data together, then how
do you plot? Well, the first thing you do,

and I'm using this influential thing, as an
example, is you draw a horizontal line and

you make that your x axis, now you gathered
your data based on years or weeks or something.

So you can label those time periods there,
because you already know those time periods.

And so you just label that x axis. There,
then you draw the vertical line for your y

axis. And again, you've done all your measurements,
right? So if you were measuring how many times

you went to the gym per week, you know, maybe
once a day, you know, that would be seven

would be the maximum, right? So you didn't
want to make sure your y axis is tall enough

to get that seven. And if you had a good week
there. And so that's really what you're looking

for in the y axis, you don't want to too tall,
like you see the highest point that they have

Ooh, in 2009, they had an outbreak there,
they needed to make sure that the y axis was

tall enough so that they could graph that.
But other than that, you don't want to too

much taller. And then make sure you label
it. I'm big on labeling here, because otherwise

people get confused.

Okay, now we're going on to the next step,
then this is where you get into actually putting

in your data. Now, because there were so many
weeks, like if you look at like 2007 is only

about like the x axis is only about two inches
wide. And all like 52 weeks of 2007 were plotted

in there. So it literally looks like a super
smooth line. But honestly, what they did was

they went and they put each point in. And
so they put each point in separately, and

then they connected the dots. And that's why
it looks so smooth. If you only have

a few

points, and you have a wider x axis, it'll
be a more choppier, it will be, it'll look

a little bit more like they'll stock market.
Graphs like that go up and down, up and down

and kind of look like a roller coaster and
not so smooth. But if you have a lot of points

and you mission together ends up looking really
smooth. You also I just wanted to point out

can have more than one line on the graph.
For more than one set of data values. Like

here, they're comparing, I don't know some
sort of book performance, how much it was

sold. In US versus Canada, you just have to
make sure that you have a legend if you do

that, so people can tell the lines apart.
So to summarize, time series graphs are useful

for understanding trends over time, like whether
things go up or down like you saw on that

influenza chart, we could see when there apparently
was kind of an epidemic or an outbreak. So

graphing more than one set of time series
data, like you saw in the last graph on one

graph can help and comparing the differences
between the datasets I worked at for the US

Army. And there's a lot of problems with people
getting injured in the army. And so I made

a lot of time series graphs of rates of injury
over the years because we were trying to do

things to make the rates of injury go down.
And then that way we could see if the trend

was there that we were actually making them
go down. So that's the main goal of these

time series graphs. Now, I'm going to move
on to talk about a bar graph, which can display

quantitative or qualitative data. And I'm
going to first start with the features of

the bar graph. here's just an example on the
right here. I'm going to talk about how to

make one and then we're going to talk about
what happens when you change the scale meaning

the x axis like how how tall the x axis is,
on a bar chart because it really changes things.

I call it a bar chart sometimes, or bar graph.
They're really the same thing. I don't know

why they chose graph in the book. But then
finally, there's I want to do A little shout

out to what purrito charts are, we don't really
use them much in healthcare, but I still wanted

you to know about them. Alright, so let's
look at the features of a bar graph. The first

thing you want to know is that they the bars
can be vertical or horizontal. So don't, even

though I'm showing you this horizontal, or
this vertical example, don't be thrown off,

if you see a horizontal example. Regardless
of whether they're vertical or horizontal,

the bars are supposed to have a uniform width,
and uniform spacing, they can't be wider or

skinnier. And they have to be spaced apart
at a uniform rate. I'm gonna use, like I said,

this big one here, as an example, to talk
about bar graphs, I just want you to notice

what is being graphed here. And this is the
percentage of people in the US not covered

by health insurance. And it's split up by
race and ethnicity. And it's looking at the

years 2008 through 2012, which is like bad,
right? Like, you want people to have health

insurance. Okay, um, so item three here says
the length of the bars represent either the

variables frequency or percentage of occurrence.
So if we were looking at instead of percent

like it's I've circled percentage, because
that's what we're looking at in this one,

we could have looked at, you know, number
of visits at a health care clinic, and that

would be frequency, right. But we haven't
been looking at percentage here. So I, so

I just wanted to call that out. So you'll
see then, on the y axis, we have the measurement

scale. And as long as we write it there, and
we use that same measurement scale, for graphing

each of the bars, we will be fulfilling the
item for which is the same measurement scale

is used for each mark. I don't know why anybody
do it any other way. But that's part of the

features of the bar graph. Now, this is a
feature that really is like my pet peeve,

I get so irritated when I find a bar graph
or any other graph where things are not labeled,

I get totally confused. So you really want
to put on a title, you need to put the bar

labels, at least on the app on the x axis,
right? Like you have to know see how it says

white alone, black alone, like you wouldn't
even know what those bars were unless somebody

put something there, right. And some people
also add the actual values for each bar, I'll

do that if there's space, like there was space
here. If it gets too busy, I don't do that.

But um, because you can kind of see them from
the graph.

Now, you're probably wondering, um, you're
probably kind of having a flashback, you're

like, this looks totally like a histogram.
What is the difference? Well, I started by

talking to you about histograms, they're actually
a special case of a bar graph, right? So bar

graphs are more general. And the histogram
is a specific type of bar graph. So histograms

are bar graphs that must have classes of a
quantitative variable on the x axis. So you

can already see that the bar graph I'm showing
you is not a histogram, because it says categorical,

qualitative things, it doesn't have a class,
right? Also histograms must have frequency

or relative frequency on the y axis, which
as you can see this as percentage of something.

So that's not that. So this isn't a histogram.
But whenever you make a histogram, you're

just making kind of a special bar graph. And
I just wanted to point that out, so you weren't

confused. Now, I said, I was going to warn
you about what goes wrong when you change

the scale. And what I mean by changing the
scale is when you look at that y axis, notice

how it the top of it the way this person made,
it, is at 35, or 35%. But notice that the

highest racial group without health insurance,
which is unfortunately, those of Hispanic

origin, that that's close to 30. But it's
not all the way up to 35. So I'm not exactly

sure why they made it so high. So I wanted
to see what would happen, what the shape would

change these bars, if I actually made the
top 30. So I regenerated this, and then you'll

see what happens. See, it's the same data.
I just made it and I made the top 30. It's

kind of subtle, but suddenly all the bars
look bigger, right? So if I were like some

advocate and running around saying this is
terrible, you know, these people don't have

insurance. I'd like to look at the one on
the left more than the one on the right. But,

you know, in a way, that's a little misleading,
right? It's the same data. So the differences

between bars are more dramatic when we change
the scale to be shorter, a little bit more

dramatic. But let's go The other way, and
this is where I see people do things a lot.

Let's see what happens, see how that the the
top of the y axis is 35. Right now, let's

double that. Let's just make it 70. And then
let's see what happens. As you can see, the

differences between the bars look small, right?
Like, the difference between that big Hispanic

origin one and the lower white and Asian alone
ones isn't really that big anymore. So my

opponents would rather look at that graph.
In fact, everything looks kind of small. on

that graph, it's a Oh, there's no problems
with insurance. Um, and that's, you know,

when people talk about lying with statistics,
so to speak, I mean, these are the kind of

tricks people do to try and change how things
appear. And the best way to do it is to just

do kind of what I suggested is look at the
next one up from your tallest one. And do

that, use that as your top of your y axis,
what I would have to do with the army is I

was looking at rate of knee injury, and also
rate of ankle injury. But knee injury was

way more common. And so if I wanted to compare
the two, I always use the same scale, because

otherwise, people wouldn't be able to see
that the ankle injury was really, really low.

Compared to the knee injury, even though they're
both important. Um, let's hall with a taller

y axis, the differences between the bars look
dress less dramatic, and also the taller you

make your y axis, the less it looks like you
have of the bars, so you got to be really

careful. I don't think you would do that.
But you know, other people do that, when they're

trying to make their points. So just be careful
for that. Also, a term that was mentioned

in the book is the term clustered and clustered
bar graph. It's not that complicated, it just

means more than one bar is graph for each
category. You'll see in the in the last one

I did, it was just on on one topic. And here,
if you look at this one on the right, and

of course, I mixed it up a little I did the
horizontal version. But this is life expectancy

at birth.

And it's it's separated by you'll see that
there's three sets of bars, right? There's

both sexes together, in there's a bunch of
bars for that. And you see the legend Hispanic,

non Hispanic, black, non Hispanic, white,
and then they mix them all together all races

origin. And then they also have separate set
of bars for male and female. And so this would

be clustered. And if you do that, you really
need a legend so people can tell what's going

on. You'll also notice that you know, life
expectancy, that's good. If it's high, right,

you want to live to be 8090 100. But if you
look at the bottom of the slide where we have

the x axis, if we mean if we started at zero,
and just made it all long, it would not even

fit on the slide. So what they'll do is they'll
make these little hash marks with this little

squiggle, and indicate that they just skipped
ahead. But like I said in the first part of

this, if they skip ahead on the female one,
they have to skip ahead on all of them. Right,

so everything is skipped ahead there. This
is a fair comparison. It's just like we're

sort of, it's like, we're fast forwarding
through the movie up to about 50. And then

looking at the differences there because everything's
the same up to that. So that's just another

thing about scale is notice whether it's clustered
if you've got a legend, and also look for

the squiggle. Okay, now I'm going to give
you a shout out to a purrito chart. And you

probably already noticed, we don't really
use these much in healthcare, because this

example is about causes of an engine overheating.
Well, we don't do that a lot in healthcare.

And you'll see I kind of slapped on a label
on the y axis, the word frequency, okay. So

in a perrito chart, this is you remember how
I was saying this histogram is a special bar

chart, or bar graph will pre though chart
is a different kind of special bar graph.

Okay. And then that one, the height of the
bar indicates the frequency of an event. Like

if you look at these events here, like damage
radiator core, that happened 31 times right?

And then happened more often than faulty fans,
which only happened 20 times. So what they

do is they figure out what happened the most
and the second most and least whatever, and

they deliberately arranged them in order left
to right, according to decreasing height.

It's a way of sort of zoning in on what is
the most important problem you're finding.

So it's really meant to graph frequencies
of problems. I actually only saw one purrito

chart I've ever ever in healthcare, so So
far, I really looked for one. And what it

was about was, it was about things that can
happen that are bad in a nursing home. And

I remember the tallest bar was for falls,
right? Like people fall in a nursing home.

And then there was a smaller bar for medication
errors that happens. The reason why we don't

I think the reason why we don't use these
a lot in healthcare is, you know, let's pretend

that's what this was of it, let's pretend
this 31 instead of damage radiator course

that 31 Falls? Well, the first thing you'd
probably ask is, well, how many people are

in that, that nursing home? You know, and
how long did you collect data for right? 31

Falls is pretty bad. But it's not bad. If
you have hundreds of people over 10 years

of that all you get a 31 Falls, you're doing
pretty well. So I would say that the reason

why we don't use preto charts a lot in healthcare
is that sort of leaves out some important

information about these serious events. And
so we like to look at things in different

ways. So just to summarize, about bar graphs,
bar graphs must be made following a few rules,

I talked to you about the you know the difference.
with, you know, you have to keep the width

the same and, and how you have to label the
axes. So we know what you're talking about.

Because you can visualize both quantitative
and qualitative data using a bar chart. So

these labels become really important, as do
scales, right? Like, I showed you how you

change the scale, and you can make things
look different. So you want to be careful

and be cognizant of that. And also, I did
a shout out to purrito charts, and I explained

why I think they're not used that much in
healthcare.

Now we're going to jump into pie charts. You
know, just even the thought of a pie chart

makes me hungry, doesn't make you hungry.
Um, so here's what a pie chart is. They're

also called circle graphs. They're used with
counts or frequencies that are mutually exclusive.

And that sounds really fancy. But all it means
is when every individual can only fall in

one category. So I'm going to give you the
example on the right, which is actually from

a real report you should probably read. It
was a survey that was done by the Massachusetts

nursing Association, and they got 339 nurses
to fill out the survey, one of the questions

was, do you receive annual blood borne pathogen
training? Now the answer is only going to

be yes or no. They can't say yes and no. That
is what mutually exclusive is, is where you

can only answer one answer. So as you can
see, 234 people said yes, which is good. And

105 said no, which is bad, I'm worried about
that.

But these pie charts

are often made in graphing programs, because
they're a little difficult to do by hand.

And I'll explain to you why. And unlike peredo,
charts, these are super common in healthcare,

as you can see right there on the slide. So
let's look at the features of a pie chart.

Um, I actually just made up this fake pie
chart, I pretended I had a class where I gave

a five point quiz, right? And the reason why
I did that is I wanted to show you how to

do it with a quantitative variable. Because
remember, the last one, it was yes or no.

And that's qualitative. Those are the the
answers that the nurses could give to that

survey question. Well, this is a different
one. This is where I actually put, you know,

fake students in their their points on this
quiz into classes, right? Like you see zero

points, one to two points, three to four points
and five points, right. So regardless of whether

you're doing yes, no, no qualitative, or,
you know, different categories like that,

or you're doing classes like this, every individual
in your data must be in only one of the categories,

only one of the classes kind of like frequency
tables and histograms. You everybody gets

one vote. And that's really important in a
pie chart, even though it can be used with

qualitative or quantitative variables. And
you'll see later What I mean by that. And

so here is just a fake example I made of how
you would then make a pie chart out of a quantitative

variable.

So

I'm just gonna briefly go over how you would
do this by hand and I'm realizing I've never

done this by hand. I always use Excel as you
probably recognize that lovely purple color,

which comes out of Excel. But if you were
going to do it by hand, I guess you'd have

to go buy one of those things in the lower
left, which is a protractor because that helps

you see the degrees of a circle. Remember,
it's a whole circle has 360 degrees, right?

I don't know if you remember all this from
like trigonometry. And but then like a half

circle would be 182 Freeze. And so that's
how you figure out like how much of the piece

of the pie you need is using this protractor.
So if you're going to make a pie chart by

hand, you first have to make a table, you'll
see we make tables constantly and statistics.

And I put class in the first column, because
I was doing one that required class because

it's quantitative. If you were doing that
one with the nurses saying yes or no, you

would put category and you just say yes or
no, right, and then total, then of course,

next, you put the frequency. And I always
put total to add it up to try and make sure

you know my fake class apparently, and 37
people in it. So I just want to make sure

you know, everything adds up, then the next
step room will remind you of relative frequency,

it's where you figure out the proportion of
the circle that that's going to take up, right.

So see, the five points out the seven people
who got five points? Well, if you divide seven

by 37, you're going to get point one, nine,
well, that's I like percent. So that's 19%.

So that would say what proportions the circle
they get, right. And then finally, in the

last column, remember how it's telling you
the whole circle is 360 degrees, when you

take that proportion you get, and you multiply
it by 360, to figure out how many degrees,

you're going to make your circle. And that's
why you need the protractor. And that's also

why I always use Excel for this because it
makes it so you don't have to worry about

those things. All you would need for Excel
is actually just the class or the categories,

and the frequency. And then if you use their
automatic pie graph function, then you can

get all this other stuff out very quickly.
So I just wanted to make a few notes about

pie charts. This is the thing I'm coming back
to is this mutually exclusive categories.

So I want you to imagine that I do a survey,
right. And I asked the question, what is your

favorite color? And I give some choices like
red, green, blue, whatever, there's only going

to be one answer to everybody's question,
right? Because you can only have one favorite,

right? And that then is eligible to be used
in a pie chart, because everybody gets one

vote. But a lot of times, I'll see people
who do a different survey question, they'll

say, check off all of the colors you like.
So if I get that I'm like, Oh, I love red.

I like orange, I like green, I'm checking
off a bunch. There's some people I know who

don't really like color, like they just were
gray and black. So they probably wouldn't

check off anything. And then there are the
people who just check off one or two. Well,

as you can see, people can have multiple votes
or no votes or whatever. And if you have that

situation, like I was telling you, where people
can say multiple things, you've got to go

into bargraph land, okay? Because a whole
bunch of people can like read a whole bunch

of people can like green, a whole bunch of
people can like blue. And you won't get a

circle out of that. If everybody answers just
one answer. And so therefore, everybody's

in a mutually exclusive category, then you
can use the pie chart. I also wanted to let

you know that I find it and I think a lot
of people do more informative to put the percentage

on the actual chart, then the frequency, some
people put both the frequency and the percentage,

which is good, it's not so helpful to just
put the frequency as you see that the nursing

report did on the left. And it's because you
really don't know, you know, 234 seems like

a lot. But what proportion is that of the
circle, that's what you would kind of want

to know. Whereas if you look on the right
on mine, you can see like, for instance, only

5% God zero point, that's a small amount,
right? You know what 5% means? It's just hard

to tell, you know, if you look at that one
on the left, and looks a little like two thirds,

which would be 66%. But we don't know what
the percent is, right. And so it's really

helpful to have that percent. And always include
a title and a legend. Because if you're, if

you're graphing a pie chart, you're gonna
have more than one category, and so people

are gonna want to know what that color means.

This looks so good, doesn't look good. Um,
pie charts are common in healthcare, and they

graph mutually exclusive categories. Okay,
so so you'll see this all the time. And like

I said, it's easier to make using software,
I use Excel, it can come out of other software,

but I just like Excel because you can really
put fancy labels on and you can do that squiggle

thing and but choosing a graph requires some
consideration, like whether or not you actually

want to make a pie chart or a bar chart or
whatever, requires some thought. And also,

regardless of the chart you make, you should
follow these rules. You should always provide

a title, okay? Even if it's just for your
private use. Trust me, I've done this. I go

back and I'm like, I don't even know what
I grabbed. So take your time sit down, write

a little title. So you remember what you also
labeled the axes. Because, again, you think

you're going to remember or maybe you think
it's obvious everybody in the audience is

going to tell, don't leave anything to be
assumed, just be absolutely clear about what's

on each axis. Always identify your units of
measure. So if you're talking about a rate

per 10,000 people or a percentage, or maybe
you're talking about an average, or you're

talking about a frequency, it doesn't matter,
just make sure you're clear about what you're

talking about. In the units of measure, usually,
this ends up on the y axis. So the thought

is to make the graph as clear as possible,
thinking font size, thinking number of items

graph, you know, I've sometimes seen a bunch
of time series graphs where they put so many

lines on there, I can't even see anything.
Or they'll have these really tiny font sizes.

Or they'll just try to put too much on one
graph. And it's hard to read. So if you find,

if you have trouble reading it, probably everybody
else will. So you want to modify it. So I

just throw this on the right. Can you tell
what's missing from the above graph? The above

graph is really missing a lot of information.
I mean, we don't even know what it's about

we, we can kind of guess it's a time series
graph because of the time at the bottom. But

what else right? So the person who made this
really knew what they were talking about,

but we don't, and you don't want that to happen
to your graph. Okay, so here, what I'm going

to do is review all the different graphs I've
talked about in chapter two, and talk about

the cases where that graph is useful. So you
can keep the straight in your heads what why

we have all these graphs, right. So first,
there's the frequency histogram. Remember

that that was only for quantitative data.
And that's what you make when you want to

see the distribution, right? Remember, the
distribution was a shape. And, and a frequency

histogram is a particular type of bar graph
that is meant for showing these distributions.

I also showed you how to make a relative frequency
histogram, which is almost the same thing,

only it graphs the relative frequency instead
of the frequency. And that also will show

you the distribution, right, because the pattern
will be the same. But this one's specifically

good for comparing to other data. So if you
have two sets of data, maybe from two different

locations are two different groups, then you
want to use the relative frequency histogram,

because then it's easier to compare distributions,
right. I also showed you how to make a stem

and leaf display, I explained what the stem
and leaf is, what the leaves are, and what

the stem is. And that's also for quantitative
data. And that's also if you want to see the

distribution, it's also good for organizing
the data, it's a little easier to make by

hand than a histogram. Because a histogram
makes you make a frequency table first, and

stem and leaf display, you can kind of skip
that step. So again, these first three were

just about trying to take quantitative data
and visualize it so you can look at distributions

and also look for outliers. Next, we went
into the time series graph. And that is really

about time, right? That's for graphing a variable
that changes over time. And as measured at

regular intervals, mainly to see trends like
is it going up? Is it going down? Was there

an epidemic, and that's what a time series
graph is for a bar graph. Now this is the

generic bar graph, not the specific histogram,
like I described, but the generic bar graph

can be used for qualitative data or for quantitative
data. And it can be used for displaying frequency

or percentage, and we went over some examples.
Then I shouted out to the perrito chart, which

is a special bar graph, right. And that special
bar graph graphs frequencies of rare events,

in descending order, usually bad things, you
know, rare bad things. And again, we don't

really use this much in healthcare. Finally,
I went over the pie graph. And that's four

mutually exclusive categories, quantitative
or qualitative. And we use those a lot in

healthcare.

So in conclusion, in this particular lecture,
I first went over the time series graphs,

and explained how they show changes over time.
And then I went over bar graphs and showed

you how they can display quantitative and
qualitative data. They can be up and down

or horizontal. I showed you some different
examples. And then we went through pie charts,

looking at mutually exclusive categories,
which I think are my favorite, like look at

this pie. This makes me so hungry. Um, but
at the end, it's important to pick the right

chart. Because you want to have a useful visualization
of your data. If you're trying to look for

a distribution. Choose the right kind of visualizations,
the right kind of graphs, if you want to instead

look for trends over time. You get to choose
the right kind of work. So I gave you some

pointers on how to do that. And now my mouth
is watering. So I'm gonna go eat some pie.

Yoo hoo, it's Monica wahi. Again, your statistics
lecturer from labarre College, I decided to

chop up chapter two and reconfigure it. So
this first lecture is going to be on part

of chapter 2.1, frequency tables, and the
entire chapter 2.3, which is stem and leaf

displays. So here are your learning objectives
for this lecture. At the end of this lecture,

you should be able to state the steps for
making a frequency table defined class, upper

class limit and lower class limit, you should
be able to explain what relative frequency

is and why it's useful for comparing groups.
Also, you should be able to state the steps

for making a stem and leaf display. And finally,
you should be able to describe the difference

between an ordered and ordered leaf. And if
all that sounds foreign to you, don't worry,

you'll understand it all at the end of this
lecture. So just to introduce what I'm going

to cover, first, I'm going to define for you
what a frequency table actually is. And then

I'll explain to you how to make one which
will help you understand even better what

it is. After that I'm jumping right into what
a stem and leaf display is, and how to make

one of those in the main reason why I can
combine these is because I feel like stem

and leaf displays can help you make frequency
tables. That connection was not really made

in the book. So I'm making it here. So let's
just start with the frequency table. So what

is one of those? Well, you know, when I think
of frequency, I think of the radio, right?

Like I think of REM what's the frequency?
KENNETH? I think that was a last hit. Okay,

that's not what we're talking about. We're
talking about frequency, like the word frequently,

like How frequently do you go to work per
week, right. And you would count how many

times you go to work or go to class per week?

Well, frequency

is, like frequently, it's like how frequent
something happens. So first, I'm going to

explain to you what a frequency table is,
and why you make them, then I'm going to define

some more terms, I just defined frequency,
I'm going to just define some more that you're

going to need to know. And then I'm going
to explain the steps for making a frequency

table and a relative frequency table. So remember,
quantitative data, I'll just remind you qualitative

data are categorical. So that's like gender
race diagnosis, where you put individuals

into categories. And quantitative data are
numerical. Remember, like age, heart rate,

blood pressure. Now, I just want to calibrate
you to the idea that this whole frequency

table thing, this, this whole thing is about
quantitative data. And so this entire lecture

actually is focusing only on quantitative
data and not qualitative data already. So

when you have quantitative data, as you probably
noticed, if you've ever had it, right, like,

let's say that you, let's say you go on Yelp,
you know, I always give that example. And

you tried to decide whether to go to a restaurant
or not. You have a bunch of fives, and fours

and threes and twos and one stars, how do
you know, you know, you just have a pile of

numbers. So how do you organize them, I'm
going to give you like a totally fake example

I made up Okay, so I'm pretending that 60
patients were studied for the distance, they

needed to be transported in an ambulance.
So how far they needed to be transported from

where they call the ambulance, and were picked
up and actually got to the hospital. So the

shortest transport in my fake data, or the
minimum was one mile, which is awesome. That's

kind of what happens to me because I live
right near a hospital, hopefully, I don't

need to be in an ambulance very often. But
that's what happens in urban centers, the

longest transport the maximum was 47 miles,
which would really suck. And I just want to

point that out that happens to people in the
rural areas because of lack of access. So

this is kind of realistic, even though it's
fake data. But anyway, it's hard to just look

at a pile of numbers. So how do we understand
these data? Well, now I'm going to start those

definitions. The word class means the interval
in the data. So in Remember, we're talking

quantitative data. So let's say I just made
up well, how many people got transported between

30 and 40 miles, okay. That would be a class
of 30 to 40, right. And the class limit is

the lowest and highest value that can fit
in the class. So carrying on with my example

of a class I just randomly picked 30 to 40.
If we made that a class we would say 30 would

be the lower class. limit, and 40 would be
the upper class limit. Make sense? Alrighty.

So then, of course, you have the width of
the class or the class width. So that's how

wide the classes. So carrying on with the
example, if the upper class limit was 40,

and the lower class limit was 30, what you
do is you minus 30, from 40, which you get

10. And then you add one, and n equals 11.
That's a little formula. But if you're like

me, and you count on your fingers, you would
go 3031 32 6034, blah, blah, blah, and you'd

realize that there are 11 numbers in that.
Now we get to frequency, like I sort of quickly

explained in that is how many values from
the data fall in the class. So how many patients

were transported 30 to 40 miles. Or another
way of saying it is, if you look in all the

data you have, and you find every single person
that either got 3031 3233, blah, blah, blah,

up to 40, count all those people up that then
you will get the frequency for that class.

Okay, but you probably realize you do need
to decide on classes before you go counting

frequencies, because you need to know the
lower and upper class limits. So let's talk

about some rules about classes. First of all,
classes have to be the same width, you can

have 30 to 40, and then 40 to 42, right, or
41 to 42, right? You can't have skinny class,

fat class, they have to have the same width.
But, um, there are different ways to pick

it, right? So, class width can be determined
empirically isn't that a fancy word empirically

just means you just choose it because you
like it, right. And if you ever look at survey

data, about just about anything, when they

look at the quantitative variable of age,
they often put that in classes. And as you'll

see on the slide, these are the classes we
often see 18 to 2425 to 3435 to 44. And you

can go on, right, like, that's what you normally
see. And that means, empirically, you just

picked it out of the hat. And already, you're
probably noticing Well, 18 to 25, or 18, to

2465. and

older, those classes

aren't really equal as the ones in the middle,
right? Like, what's the upper class limit

for 65 and older? Okay, well, that's just
normally what happens in the world, and especially

in healthcare, and healthcare, when you pick
classes. Even though the classes are technically

supposed to be the same width, you really
should be guided by the scientific literature.

And you'll see why later, when I show you
the other videos in this chapter. It's because

you really want to be able to compare whatever
you find to whatever other people have found

before you. And therefore you don't want to
cut up your classes in different ways, or

it's hard to compare them. However, in the
book, they teach this class with formula,

so I thought I should really show you that,
too. So here's the class with formula that

I don't really see used much in healthcare
statistics, but I'm going to teach you anyway.

So this is the formula. First you calculate
this number, you find the maximum in your

data, and you're in the minimum in your data,
and you subtract the minimum from the maximum.

So the example I was giving from the fake
data about the transport is 47 was a maximum,

and the minimum was one. So I did the first
step and got 46. Okay, looking back into the

formula, you divide whatever you got there
by the number of classes desired. In other

words, like however many, you know, categories
you want, right. So if you never want too

many, like you don't want 10 or something,
you know, 34567, usually something in that

range is a good number of classes. So let's
pick six just for fun. So we'll take that

46 number we got we divided by six and we
get 7.7. Then back to the formula side, how

you decide then your class width is you increase
this number, you get to the next whole number.

Now a lot of people are confused by that,
because even if I've gotten something like

low, like 7.1, I'd still go up to eight, you
have to increase it up to the next whole number.

So you have like this, this integer, you know,
that's a number without any decimals after

it. So you have this integer for your class
with so our class with in this example then

would be eight. So, um, now I described to
you that whole class with, but I'm not going

to use it in the example because we don't
really do that much in healthcare and it makes

it actually kind of hard to understand because
you want something that's a little intuitive,

like if you look on the slide right now, you
know, less than 20 miles 21 to 2930 to 39

and then 40 or more, that may A little more
sense in your head. You know, that's how we

think of miles. If I had put like 18 to 24,
and 25 to 29, you know, we don't really think

that way. So this is helpful in healthcare
to boil it down to something like this. And

by the way, if I was writing a real paper
in the sort of real data, I'd be looking at

the papers before this that talked about transport
times and looking at those

class limits. Okay,

so a frequency table displays each class,
along with the frequency, the number of data

points in each class, as you can see, the
class limits are on the left side of the simple

frequency table, you know, the classes, and
then the frequencies on the right side, right.

And you'll notice that they all add up to
60, because we measured 60, fake patients,

and it's really good to do that little check.
Because you don't want to double count people

put them in two classes, they only get to
be in one, etc. So selecting arbitrary class

limits, can make the frequency table unbalanced.
So in other words, doing this empirical thing

can make it sort of weird because less than
20 is big, and 40 or more miles is big. And

it's bigger than the other classes. So it's
does it kind of breaks the rules of class

with but not following the scientific literature
can make your results not comparable, and

can make the science less useful. And so that's
why I sort of flail against the book with

this class with formula thing. So I'm, I'm
going to just give you another example for

a frequency table. Okay. This one is more,
it's also health carry, you know, glucose

is measured in the blood and expressed in
milligrams, 400 milliliters, right? So glucose

is a huge molecule, and it should be cleared
from the blood, especially a fasting. So if

you're not eating anything, you're not putting
any glucose in your body supposed to be like

metabolizing. That problem is some people
don't metabolize glucose very well, you know,

that's what diabetes is. So you, you care
about how much glucose is sitting around people's

blood.

So blood

glucose levels for a random sample of 70,
women were recorded after a 12 hour fast.

And this is what they got, they got the minimum
was 45, the maximum was 109. And they picked

six classes. So this is how they set up their
class limits. And again, this is using a class

with formula. And just to demonstrate, you
know, it sort of comes out a little weird

here. But then they they got these frequencies,
okay. And this is again, just another example,

using this time the class width formula to
get our six classes and to make sure that

they covered everybody. Now, you'll notice
in this, we start with the minimum like 45

to 55. And we end with the maximum, which
is up to 110. And that's really the clearest

way to do it. It's just not typically done
that way. If you read, like scientific literature

and healthcare, you just don't see these frequency
tables labeled like that. So and just to wrap

up this part, make sure all of your data points
are accounted for only once in one of the

classes. So whether you use a class with formula,
or you use empirical or arbitrarily picked

classes, every single data point only gets
one vote, it can only be in one of the classes.

And, and also, you don't want to leave any
of the data points out. So you want to make

sure that that happens that you account for
all of them. And also you need to make sure

your classes cover all the data, right. And
healthcare when we do that thing up to 20,

and 65. And over all that stuff, we cause
that to happen. However, if you're going to

use a class with formula, you really have
to pay attention to where your minimum and

your maximum are. Because then you want to
make sure all of your classes cover all of

your data. And like I mentioned, make sure
the total of your classes of the frequencies

in your classes adds up to the total number
of data points, it's just a little check,

make sure you didn't do something wrong. Now
I'm going to talk about what is a relative

frequency table. And that builds on what you
already just learned about frequency. So we

all know what our relatives are. They're like
our family, right? We have relationships with

them. And so what relative means is in relationship
to the rest of the data, okay? So in statistics,

they often use this fancy F to stand for frequency.
And, as I've mentioned before, the sample

size, if you have a sample, they use a lowercase
n. So what they use as the formula for relative

frequency is F divided by n. And if you're
clever with math, you realize what that means

is is if you take a frequency of any of the
classes, you know, it's just a portion of

the whole sample, and you divide it by the
total sample, which is that n you You'll get

the proportion of values that are in that
class, it's not really that fancy. So relative

frequency is something very useful to put
in a frequency table. So you'll see that I,

I kind of crammed it in onto the right side,
this is the old frequency table I just showed

you with glucose, but I crammed in this relative
frequency next to it. So it's super easy to

calculate, like, for example, for the first
one, see, 45 to 55, the frequency is three,

what did I do? Pull out the old calculator?
Well, I actually I use Excel. And I did three

divided by 70, because I was a total. And
I got Oh point oh four. And those of you don't

really like proportions, you can do that thing
where you move the decimal two places to the

right, and then put us percent sign. So that
would be like 4% of those 70 people are in

that first class. And then the same thing
happened with the next one, I took, you know,

the 56 to 66, I took seven divided by 70,
which came out 2.10. And those of you into

percents, I'm really into percents, I like
moving that decimal over, I think of it as

10%, then, but whatever, as you can see at
the bottom, and all has to equal 1.0. If you

like proportion, land, or 100%, if you're
like me, and you like percent land. But in

any case, this is all you have to do to do
the relative frequency table, you just make

another column and do all those calculations.
And it's super easy to calculate it. And it's

very helpful. So why did we even do this,
because we had a pile of

quantitative data, and it was really hard
to organize right. And the first thing was

we had to do was select class width. And I
talked about the politics behind that. But

ultimately, whatever you do you do in the
lower in the upper class limits need to be

determined and put in the first column of
your frequency table. Then in your second

column, which are the frequencies, you count
up, how many are in that class, and you fill

it in. And then if you make that third column,
then you can do that dividing thing and get

your relative frequencies. And that's great.
That's how you build your frequency table.

And as I go through future lectures, you'll
see even more why you would make that table

like how useful that can be. Given that you
have quantitative data, and it kind of gets

all over the place, it's very helpful to organize
it in that table.

Now I'm going to

move on to talk about the stem and leaf. And
the reason why I picked talking about it.

Now it's because it's on the theme of organizing
quantitative data. So I'm going to talk to

you about what the stem and leaf plot actually
is. Here's a just an example on the slide

and how you make one. And why why you might
make one of these you'll find it feels a lot

like making a frequency table. But why do
you make these instead of a frequency table?

And it's just more food for thought. So first,
one of the things that I got hung up on when

I took biostatistics is I could not get over
the fact that it was called a stem and leaf.

So I had to understand that. So this is an
example of a stem and leaf there. So why is

it called a seven leaf? Well, there's always
the stem. And that's so see these corn stalks,

I'm from Minnesota, I'm used to seeing them,
you'll notice that there's a stem, right,

like this big corn stock has the stem, that
thing you see that vertical line and a bunch

of numbers on the left, that part of the stem
and leaf plot is called the stem. And then

leaves are added onto the sim as we tally
up the length of the leaves. And that may

not make much sense right now, but I'll show
you how to make one. But essentially, what

you end up doing is adding these leafs like
you see under two, there's a little leaf that

just has a zero on it. But if you see under
five, there's this big long leaf with a whole

bunch of numbers off of it. So I'm making
one will help you understand this terminology.

But I first wanted to just show you this picture
because it's actually kind of hard to understand

what's going on with a stem leaf unless you
understand that that vertical line in the

numbers to the left of it is considered a
stem. And then each one of these things we

build off start, you know, off of each of
those numbers is called a leaf. So people

talk about the four leaf in the five leaf
already. Okay, so again, I'm just so into

making up data, right? So I decided to make
up data from 42 patients who visited a primary

care clinic and referred to mental health.
Now the reason why I made update on the subject

is I'm very upset about this subject. I think
people are waiting too long to get mental

health treatment. Especially if you've been
following the news about the Veterans Administration.

In the US. A lot of people are put on hold
even for primary care. You know, they're put

on waiting lists and I don't like so I made
a fake data by That as a demonstration just

to highlight these issues. Okay, so what what
data Did I make up, I made up the the number

of days between the referral and their first
mental health appointment. That was what was

collected. So let's say you go in on January
1, and you get a referral. And then 10 days

later, you actually show up at the clinic,
then that would be 10. Right? That would be

your value. So that's quantitative. So let's
take a look at it. So on the right side of

the slide, you see just this pile of numbers
from all these people that came in and, and

then got a referral. So like, you look at
the first person had to wait a

month,

go see a mental health professional. But if
you look, you know, the third one, and that

person only needed 12 days. So that's how
you sort of consume this fake data I made.

And then you'll see over on the on the left
side, I already made a step. It's blank that

doesn't have any numbers on it, but I knew
I need that vertical line. So I just made

that in preparation. Okay, so let's build
our simile.

So what we do is we start with the first number,
and that's what's awesome about this is you

just start with the first number. And if you
want, you can kind of cross them out as you

go along

to keep track. So we start with this first
number. And you'll see what I did, I went

over to the stem, and I put the three on the
left side of the stem and the zero on the

right, this begins the three leaf,

okay.

Here's the next number. Now, I put the two
above the three because it's like right before

it and you can kind of imagine we're gonna
walk down like 23456. And then I put the seven

on the right side to start the the two leaf.
Alrighty, here we are with the next number,

which is 12. And as you'll see, I started
the one leaf, you're starting to see the pattern,

right? And you can probably guess what's going
to happen next, we start the four leaf and

put the two there. Okay, our next leaf, we've
already started, right for 35. So what do

we do there? Well, we just add the five on
to the three leaf, the three leaf was already

started with that, that 30 at the beginning,
so we just pile a five on there. Here's 47,

we just pile a seven on there. Now you'll
notice I tried to line up that seven on the

four leaf with the five on the three leaf.
When you're doing this by hand, well, even

when you're not doing it by hand, you really
have to keep those things lined up or you

you won't have a good stem and leaf. Okay.
Now I'm going to just fast forward a little

a little because you can probably imagine
how to do the next row the 3836. You just

keep piling it on. But I want to show you
what happens when you get to the special case

here. Okay, well, we'll go with this 29. This
is the last thing before the special case.

So you'll notice that 38 got put in there,
see that eight, three leaf that 36 got put

in there, you know from the second row, see,
we put everything in there. And now we put

in the 29 look at that we got a three after
that. That's our next one. So where are we

gonna put that three? And I, you know, you
might think on three leaf but that's not right,

right? Because that's 30 something. So where
do you put the three? Well, some of you figured

this out, you have to add a zero onto your
step. So look at that, I put that zero there

and then we put the three in. And then you
can already guess how to do the 21. Next,

we'll just tack a one on to the to lead. But
then when we get to the next zero, we just

add a zero on to the zero we.

So you can probably figure out how to pile
up all of these. But I did want to talk to

you about something else that happens with
these stem leafs. As you go on adding to the

leaf, you got to be careful because you might
end up with a situation where you got something

big now I really feel sorry for this fake
person. 51 days for a mental health appointment

that's too long, right? But it causes us later
to have to add a five.

Now this can cause real estate problems, especially
on a piece of paper, you know, what have you

the four was right at the bottom of the paper,
right, it's kind of hard, maybe you have to

tape some paper at the bottom I have this
problem a lot. Um, you'll see here this, I

even had to move this up on the slide when
we got later to the 70 I'd add the seven leaf.

Now I just want to show you for some reason
the state of we didn't have any 60s. But you

still have to put that six leaf place or in
that that's got to be there. So even if you

know as we go on, if we're missing any leaves
in between, we just need the place are there

because that space has to be there. And here's
here's an outlier. We're gonna learn about

outliers pretty soon. This is a really long
time. 105 days this is kind of like VA status

right? But it And you'll see that and of course,
this is fake data, but unfortunately reflects

real data. You'll see when we get to 105,
not only did we skip the eight leaf and the

nine leaf, and we need to leave a space for
them, but 10 becomes the part of the stem.

So

if we went on to 200, or 300, I mean, that
would be awful. The wait that long, though,

the first two digits of it, like if we had
365, the 36 of the 365 would be the part of

the step. Alright, so I just did a little
demonstration to explain certain nuances of

the stem leaf that you might encounter in
your life. So now, I'm going to just reflect

back on the two ways that I've described in
this lecture for you to organize quantitative

data. First, I showed you how to make a frequency
table. But what you need to do with that one

is you need to set up classes and class with
and and to count the frequencies in there

a lot of there's a lot of pre processing a
lot of pre calculations, you really want to

think when you're doing this, and you don't
want to be distracted. However, if you're

trying to do a stem and leaf, you really can
do that on the fly, you don't need to set

up classes or class with, as you noticed,
we just went through the line of those pile

of numbers, and just crossed them off as we
put them onto the stemmen wave. And there

was really no need to count, you can tally
the data as you go through the list, you know,

cross it off. And it's just really quicker
to do. Of course, those of you who are pretty

clever saying, Well, basically you're forcing
in a stem and leaf everything to be in the

class of, you know, the 10s, right, you know,
the 20s and the 30s in the 40s. That's like

the two leaf, the three leaf and the poorly.
And yeah, it is kind of like a simplified

way of making those kinds of classes. But
in any case, I just wanted to alert you to

this because you might see some similarities
between the two. And I wanted to highlight

those as well as the differences. Now I'm
going to give you a few tricks here, I want

to tell you about the concept of an unordered
leaf. So an unordered leaf is what we were

making before when I was demonstrating, it's
just where the numbers are out of order in

the leaf like you'll see this two leaf it's
a seven, seven to nine. Well, if there were

an order would say 2779, right, like the two
would come first before the seventh and the

ninth. And the same with the three leaf that's
out of order, because you can see that it's

zero and five is fine, but eight doesn't come
before six

and five, right? That's no

problem to make an unordered leap. However,
after making an unordered version, you can

rewrite the stem and leaf in an ordered way.
So you see how I did that I rewrote the two

leaf and the three leaf. And now they're all
the leaves are in in order. Okay, you don't

have to be but you can do that. And if you
do that, if you make your stem only first

unordered the way I was demonstrating, then
you rewrite it into ordered, it is way easier

to count it up to make a frequency table no
matter what classes you choose. Or you can

just make each leaf a class. And then it's
super easy to make the frequency table. So

that's why I combined these two pieces of
the chapter together is because I wanted to

show you how you can use a stem and leaf to
help you make a frequency table. So a stem

leaf, it's just another way to organize quantitative
data. And it's easier to make kind of on the

fly than a frequency table because it requires
less preparation. And they can help you put

data in order before like in preparation for
a frequency table started to help you as a

first step to make sure that you can organize
everything. And at the end. Remember I keep

emphasizing your frequency table has to reflect
all your data points. And they can only be

in one class, blah, blah. Well this is one
way to make sure that happens is to first

do this pre organization using an ordered
stem and leaf. So in conclusion, frequency

tables and stem and leaf displays organize
data, they organize quantitative data. And

the stem and leaf may help you make a frequency
table. So you might want to start with that.

And the purpose of both of these things is
to reveal a thing called a distribution. And

I'm going to explain that in the next lecture.
Hello, it's Monica wahi. Again, your lecturer

from library college and we are moving on
to chapter 3.1 which is measures of central

tendency. And here are your learning objectives.
So at the end of this lecture, you should

be able to explain how to calculate the mean.
You should also be able to describe what a

mode is In say how many modes a dataset can
have, you should be able to demonstrate how

to find the median in the set of data with
odd number of values, as well as in a set

of data with an even number of values. And
you should also be able to define trim mean

and weighted average. All right, so what's
this measures of central tendency, I'm going

to explain that why we kind of call it that.
And then I'm going to talk about them, which

the three biggies are mode, median, and mean.
So I'm going to talk about those and explain

how to get those. Then, towards the end of
the lecture, I'm going to go into some special

situations. One is called the trimmed mean.
And the second is a weighted average. So let's

get started. What is the central tendency
thing? Well, if you think about quantitative

data, which that you can only do this with
quantitative data, not qualitative data. But

when you think of having a pile of numbers
like this, one of the things you want to know

is how much they tend towards the center.
Now, of course, you don't know where the center

is, until you start looking at the data. Some
data are kind of high up in the hundreds,

like systolic blood pressure. I give a five
point quiz and one of my classes, so those

numbers are low, like 12345. But then the
question becomes, do the group towards the

center of whatever list of data they're in?
Or don't they? How sort of sensory?

Are they?

You see these distributions on the slide?
You'll see, on the left, you'd probably say,

well, that looks more sensory than what's
on the right, you know, this normal distribution

on the left, and the skewed right distribution
on the right. And so intuitively, you kind

of know what I'm talking about. But what this
lecture is going to be about is how to actually

put numbers on the difference between what
you see on the left and what you see on the

right. So these are the numbers, these are
the measures of central tendency, we're going

to go over mode, median, and mean. And the
median is a little different, depending on

whether you have an odd number of values or
an even number of values. I mean, it means

the same thing, but you calculate it slightly
differently. So I'll go over that. And then

the mean, a lot of you already know what a
mean is, but there's a couple special means

we can make. One is called a trim mean, and
another is called weighted average, which

is a weighted mean, I don't know why they
chose the word average for that one, because

mean an average mean the same thing. But I'm
going to go over these things. Okay, well,

let's start with the mode. The mode is the
number in the data set that occurs the most

frequently. So I put up this little tiny data
set here of just five numbers. And it's obvious

that then five is the mode, right, because
it

repeats Once there, two fives there. But look,
I just changed one of them, I changed it to

a six. And now there's no mode.

So I just want you to know that a lot of data
sets don't even have a mode, there's just

no repeat at all in them. And that usually
happens when you have a broad range of numbers,

they can have like systolic blood pressure,
I mean, it would be kind of lucky, you just

got two people with the exact same one. But
that can happen. So don't think there's always

going to be a mode, there might not be one.
It's also possible to have more than one mode,

like look at that. So I've got six numbers
up there. And the two repeats once and the

three repeat ones. So you've got two modes,
right? But let's say that the three actually

repeated three times, then it would only be
one mode, because the three threes would Trump

the two twos,

right? So you can just imagine how confusing
this gets when you got a ton of numbers. What's

a little less confusing is, um, if you like
I said have a broad range of numbers, it would

be kind of a coincidence, if two patients
had the exact same systolic blood pressure

or platelet count, you know, like you get
a repeat in there. And then that would be

the mode. Of course, if you measure a whole
bunch of people, then eventually you're probably

going to get one. But I just wanted to say
and also, if you look at the slide all those

numbers, you'd really have to go through and
organize them and count them up and see if

there is a mode, there probably is one because
we see a lot of repeats. But then which was

the one that wins that's repeated the most?
Or are there two that are repeated the most,

and becomes kind of political when you really
do it. And it's not worth a lot of work, because

what does the mode tell you? It doesn't really
tell you much. It does tell you the most popular

answer. The word mode in French means fashion.
So like I put on the slide, you know Allah

mode, it's in fashion. So it's the one that's
most popular or the most common result, but

it's not used a lot in healthcare. And it's
actually not used very often once in a while.

I'll say, Oh, the mode. In the class for my
five point quiz was five, meaning everybody

did pretty well they mostly got a five. That
was the most popular result. But you hardly

ever have to say that. And so

remember, we learn

the words resistant, like if a measure is
resistant, you can't whack it out very easily.

Well, you can change things pretty easily
with the mode, the modes not resistant, I

even just demonstrated that on those slides,
by just changing one number, you can erase

the mode or add a mode or whatever. And so
it's not stable, it's not resistant. And those

are the kinds of things we don't really like
and healthcare, so we don't really use them.

So I'll move on to some cooler measures of
central tendency.

And here's a really cool one, which is called
the median. And it's the middle of the data.

And I'll explain that a little bit more what
we mean by the center of the data. Okay, so

remember, we're talking about quantitative
data. So you've got some pile of numbers,

it doesn't matter, you can always sort them
in order of lowest to highest. And I keep

talking about this five point quiz, I give
him my class. It's an easy quiz. And most

people get fives. But even so somebody gets
a four usually, or somebody doesn't show up

for the quiz, and they get a zero. And so
it doesn't matter, I can have 100 people in

the class, I still could put all of those
numbers in order of lowest to highest, even

if most of them were fives. Because you'll
get repeats in your data sometimes, right.

And also, sometimes you'll get outliers. Like
if I said one person maybe didn't take the

quiz and they get a zero. But everybody gets
else gets four and five is an easy quiz, well,

then that zero would be an outlier. So you
don't have to worry about that. And like I

said, you know, the data values sometimes
are almost the same, like almost everybody

gets a five on my quiz, because it's so easy.
So it doesn't matter. Even if you have these

weirdnesses in your data, you can still just
arrange them in order. And that's what we

mean by the median is the number that is halfway
up, or halfway down, right. So if I've got

100 people in my class, and I've got the zero
over here on the left, and I put all the,

you know, fours, and then the fives, you know,
I have to count up what 50, right to see where

the middle is. And it's probably going to
be in the five range, right. But that's all

we mean, we say, you'll take however many
values you have, put them in order, even if

there's repeats and outliers or whatever,
just put them in order, and then count up

halfway. And that's where the median is going
to be. So I'll demonstrate this here. So how

to find the median, the first step is to order
the data from the smallest to largest. So

I'm giving you two demonstrations. And I don't
even know what these data mean, I just totally

made them up. The one at the top, the data
set the top that starts with 42, that only

has five numbers in it. So I'm going to demonstrate
the odd version with them.

The one

set at the bottom has actually six numbers
in it. So I'm going to demonstrate the even

version, because remember, it goes a little
differently, whether you have an odd number

of numbers or an even number of numbers. Okay,
so those are the numbers. And we still have

to do the first step, which is order the data
from smallest to largest, because you can

see they're not in order. So I'm going to
do that here. Okay, there it is. So those

are the same numbers, they're just in order
from smallest to largest, okay. So we're going

to get rid of those numbers on the top, and
instead put the position they're in. So let's

look at the top data set, which is the odd
one. So I'm going to say this is how you find

the median is you

number the positions,

you know, it's 12345. And it's the middle
position. So you can imagine, if we had had

seven data points, we'd go out 1234. And we'd
circle that one, and that would be the median.

So that's what you have to do is you take
these, if you have odd values, you just put

them in order, and see I numbered them for
you. And then you take the middle number,

and that's the median. That's what it is.
It's 42 in this one. Okay, we'll do the downstairs

data set there that has six, as you can see,
the positions are numbered. And then what

do you do, you go to the third and fourth
position, which is the kind of the middle

right, and you literally make an average of
them, you add the two, and they happen to

be seven and eight right next to each other.
But if they had been like eight and 10, then

the average would have been nine, and that
would have been the median. But because this

is seven and eight, you do seven plus eight,
divided by two, and it's 7.5. So when you

do the median with an odd number of values,
you're going to be taking one of the values

in there. If you're doing the median, on an
even number of values, you might get something

with like a decimal, because you're looking
for the two values that straddle the middle,

and you're going to be making an average of
them. And so you might get kind of a wacky

number like 7.5 that's not in the underlying
data set. So um, this is fine for like, if

you have five or six numbers or seven. What
What if you have like 150 numbers, I mean,

you do still have to put them all in order
to begin with, you know, like I use Excel,

I probably just soared. But you have to know
how many numbers to go up. It's not obvious.

So this is how you find the middle number.
They

have a little

formula for it. So let's say we have an odd
number of values. And I'm giving you the example

like 21 love Let's say at 21 students in my
class, and that's how many values I have.

And I wanted to make a median of their grade,
what I would do is put them all in order.

And I'd say, Well, I have to go up so many,
and that's the median. But I don't know how

many to go up. So I would use this calculation.
So I take the end, which in our case is 21.

And I'd add it to one to it. And then we get
22. And then I divide by two. So that's just

how it works. So if you had 41, you would
do 41 plus one, it would be 42, divided by

two. Or if you had, like, I don't know why
I'm picking on ones like 27, you do 27 plus

one, and that would be 28. And 28 divided
by two is 14. And so you see, it would just

force it to be an even number that you come
out with. And then that's the position you

got go often. So if I had 21 students in my
class, and I took the grades and raised them

in order from lowest, lowest to highest, like
if they were that quiz grades, you know, most

of them would probably be four and five, but
it wouldn't matter, what I would do is just

start with the lowest and count up to the
11th 1/11 position, and then that would be

my meaning. Now, you also have to do that,
you have to find the middle number, even if

you have an even number of values. So I took
an example 14, now you'll notice we use the

same formula. But if you do use this formula,
you get 7.5. And that doesn't, that's not

the median. That's just how many positions
you have to go up. Right. And so remember,

on the earlier slide, we had, we had to go
between the third and fourth position, we

had to average those two numbers. Well, this
is basically saying, if you get 7.5, you have

to go to the seventh and the eighth, the one
that straddles it, and those are the two that

you average. So if my n like 100 is a nice,
even number. So if you have 100 plus one and

you get 101, then you've got, you know, 50.5,
right, and that just is a secret message that

when you line up all your data, you take the
50th, one in the row and the 51st, one in

the row, add them together, divide by two,
and that's going to be your median. So I just

wanted to share with you this little formula,
just in case, you get like a large number

of numbers thrown at you and putting them
in order is a big pain. And then you have

to figure out how many to count up, you can
use this formula to get the middle number.

So what does a median tell you, we have a
lot more to talk about here. First of all,

it's called the 50th percentile of the data,
what it means is 50%, or half of the data

points are below the median, and the other
half are above. And that intuitively makes

sense because you just created we created
this median together. And we could see that

half of the points are in the bottom half
on the top. And so it's also known as a middle

rank of the data. And what's nice about the
median is it doesn't really care much about

the ends of the data. Like if I gave extra
credit to a few people in my five point quiz,

and they got a few sixes, probably the median
won't even change because it's in the middle

where all the action is where we find the
median. And outliers don't really bother it

because like if one or two people get a zero
on the quiz, it's really, you know, if there's

21 people in there, or 100 people in there,
it really isn't gonna affect, you know, these

things happening at the end. So we like the
median because it's very resistant, and it's

very stable, you can't really whack it out
with some outliers, throwing them on the ends.

Now I'm moving on to the third measure of
central tendency, which is a mean, but I also

threw in here, trimmed mean and weighted average
because there are other kinds of means. And

we're going to talk a little bit also about
resistant measures, because like I just mentioned

that.

But I'm gonna step back

and talk a little bit about the Greek letter
sigma here, that's actually capital sigma,

I do not speak Greek. And I actually have
trouble speaking statistics, because a lot

of it's in Greek. So I try to avoid that and
my lectures, but sometimes you can't get away

from it. So I have to really introduce you
to this capital sigma. So in English, we say

or statistics ease, I guess, is whenever you
see this, you say some of Wah, like you expect

something to be right after it. Okay. So if
you see, like the sigma and then x, you would

say sum of X. That's how you say. So what
is x? Well, remember how we were just making

medians. And we were looking at modes, well,
each value there is considered an X, okay,

so each of the values in those days sets an
X. So sum of X would mean add these all up

or add up all the axes. And then I just threw
on another example, let's say somebody came

to you and said sum of X, Y, it would mean
you must have some x y's lying around and

you have to add them together. Or somebody
came up to you and said, you know, some of

the prices on your, of the food in your

basket and the grocery store, right? Somebody
said some of that, you'd be like, Okay, I

have to go through all these prices and add
them up. Right. So that's what some of them

Okay, and it's used a lot in statistics, and
we're going to use some of all the time. So

I just want you to get in your head that whenever
you see some of, there's probably going to

be this thing next to it. And it's gonna be
a batch of numbers that you have to add up.

And if it's numbers from our data set, it
will be called x, if it's other numbers from

something else that will be called whatever
they're called. But just know that this means

some of and I see on the slide, the upper
one is Times New Roman, and the lower ones

Arial, they look kind of different. But I
just wanted you to get ready to deal with

this some of a lot. Okay, so here we are,
I'm hitting you with a sum up. This is the

formula for the mean. And a lot of you already
know how to calculate the mean. And you just

kind of do it. And you didn't know this is
how you say it in statistics. But basically,

it's this ratio. So this is like a fraction.
And on the top of the fraction is a sum of

X, you add up all your actions. And on the
bottom of the fraction is an, which is however

many you have. So you add them all up and
divide by however many you have. And you've

probably been doing this your whole life.
But this is actually the formula. So I just

thought I'd demonstrated, um, see, I put that
sum of remember those six data points I was

using for the median, I just kind of copied
them over here, I add them all up. And so

I got some of axes 40, right. And then I counted
them, and that was six, while I made them

be six. And so 40 divided by six is 6.7. So
that would be the mean for these data. And

you probably already knew how to do that.
But I wanted to sort of crosshatch it with

the actual formula. Okay, now I'm again, going
to take a little break here to just talk about

means, because remember, we talked about sample
statistics and population parameters. If somebody

just talks about a mean to you, and they say,
look, the mean such and such as six or something,

unless you really get into it with them, you're
not going to tell it's not going to be obvious

if they did a sample mean, or did a population
mean? So but when we write this down, it becomes

obvious. If I say, x bar, see that x without
line above it, that's pronounced x bar, and

you'll see I write it on the sides x bar,
because it's so hard to put that little line

up there. But that means the same thing, this
x bar, whenever here x bar, or you see that

x with a line over it, it means that it's
the sample statistics. So if you ever saw

like x bar equals six, not only do you know
the mean is six, but the secret code says

this mean comes from a sample, because x bar
is being stated. But if you look on the right

side, you'll see that it says there's this
m, and it's pronounced mu, it's a Greek letter

again, and I you'll show, you'll see on the
left, I put it in Arial. And on the right,

it's n times new roman looks a little different.
But it's pronounced mu. And so if you saw

mu equal sex, you'd be like, Whoa, that was
a population they measured. And the you probably

say that too, because you don't see mu a lot
like people usually don't

measure the population,

it's a lot of work, you often see x bar, but
even so I want you to be cognizant of whether

it says mute or whether it says x bar, because
it's still going to be a mean. But if it's

mu, they're talking about the population.
And if it's x bar, they're talking about a

sample. And that might be more important later.
But just keep this in mind. Also, when we

talk about samples, we use a lowercase n to
mean the number of numbers we have. Whereas

if we use, we're talking about populations,
we use an uppercase n a capital N. So you'll

see that the sample mean formula on the left
side, this x bar equals sum of x divided by

n, it changes if you're talking about the
population mean, and you're like, come on,

you add it up the same way. Like mu is basically
the population mean, and capital and it's

just the number in the population, that means
almost the same formula. But the issue is

you really are supposed to label things what
they are. So if you're doing a population

mean, mean, you're supposed to call it mu,
and you're supposed to use, you know, write

it like that on the right side of the slide.
And if you're doing a sample mean, you're

supposed to call it x bar, and you're supposed
to do it like on the left side of the slide.

So I just wanted to make that clear to you
as you go through the rest of these lectures.

Because when I say mu, I'm gonna mean a mean,
but it's gonna be from a population. And when

I say x bar, the mean the mean, but it's gonna
mean it's from a sample. Alright, so now we've

talked about several measures of central tendency,
but I wanted to put a means and medians together

in kind of a cage match because I wanted you
to look at them and see what their differences

are. Now, I've been sort of giving accolades
to the median, right, because it is very resistant

to outliers, and it's very stable. Remember
how I pointed out if you throw some outliers

on either side, it doesn't really affect it
much. Unfortunately, means are not resistant

to outliers. You could just throw like if
I took my five point quiz, and I just felt

like failure. barring a student and then giving
them 10 points, it would totally screw up

the mean for that class. And it's so it's
not very stable. So one of the things we can

do if we've got outliers in our data is to
just use the median. But sometimes we want

to use the mean. So we got to do different
things with it. So one of the things we can

do to try and make a more stable mean, or
honest mean is to trim it. So I'm going to

talk about how you do that. So as you can
see, on the left side of the slide, a very

high value, a very low low value, like an
outlier, or more than one outlier can really

throw off the mean. And it's not a problem
with median. So if you want to make the meal

a little resistant, what you can do is trim
data off of each end. So the outliers get

cut

off,

okay? The problem is, you can't look at the
data, when you're doing that, really, you

would just have to make a rule when you're
not looking and say, Okay, I'm going to trim

X amount off the top and X amount at the bottom
and as to be equal, and you just have to look

away when you're doing. Okay, so what I'm
some people do is a 5%, trim mean, which means

you take 5% of the data at the top and cut
it off, and 5% at the bottom and cut it off.

So you basically lose 10% of your data. And
in health care, a lot of people get mad about

that they don't want to lose any data. So
they don't like to use this way of fixing

the problem of outliers, they use other ways.
But I wanted to show you this as a simple

way to fix it. So I'm going to imagine we
have 100 data points, because it just makes

it easier for you to see what's going on.
Um, so if you had 100 data points, 5% of them

would be five. So basically, you'd be trimming
five off of the top, and five off the bottom.

So the first step would be is probably you
already made the mean out of this 100. And

you didn't like it because you saw outliers
at the top and bottom. So what you have to

do is put the data in order just like you
do for the median, you put them all in order,

you sort order from, you know, the lowest
to the highest, take all of your 100 and do

that, then what you would do is you would
like circle the five most bottom ones, and

they're going to get cut off, and you'd circle
the five top most one of them, they're going

to get cut off, they get thrown out. And then
you're you've got the 90 values left in the

middle. Now you make a mean out of those.
And then that's a 5% trim mean, and you got

to tell people, if you do that, you can say
here's the original mean, and here's the 5%

trimmed mean, because then people get an idea
that there must have been some outliers and

some of your data got hacked off. But then
this might give you sort of a more stable

estimate of the mean. Now I'm going to move
to something else entirely. It's not about

trying to make the mean stable, it's just
about trying to make the mean a little different.

Sometimes certain values in your mean should
count more than others towards the mean. And

that sounds really esoteric, but the way we
see it all the time is in school. So you might

get a great grade on your homework, you might
get A's on your homework, right? But if homeworks

only worth 10% of your final grade, it doesn't
help you much. And so what that 10% is it

when you have a class like that is it's called
a weight. When you move into statistics, you

say well, I'm going to, you know, I as the
teacher, I'm going to wait your homework grade

at 10% of your final grade. So it doesn't
matter how awesome your homework grade is,

or how bad it is, it's really only going to
count for 10% of your final grade. And that's

why we do weighted averages, you know, I don't
think your homework should be worth like 50%

of your grade, right? That doesn't make any
sense. And so even though, so you might want

to have different things contribute a different
amounts of weight to that final mean. So this

is a way of messing around with the mean,
and making certain things going into it count

for more, or have kind of a bigger vote than
the other ones. And so I again, I'm just gonna

stick with school to give examples because
this is where we normally see it. So I mean,

if this example where homework is worth 10%
of your final grade and quizzes would be worth

20%. And the final worth 70%. And I just want
to point out, I've actually seen people do

this, like cuz I tutor, and like this is horrible
making your final worth, like, over 50% of

your grade. So this is just a shout out to
any like professors watching this. Don't do

this. Okay. But anyway, let's say I was mean
and I did it. And let's say you were pretty

good student and you got an A on the homework,
right? And so we're gonna say that's a 4.0

because a lot of schools would say A's 4.0.
Then let's say you got B plus on the quizzes,

maybe because the lectures weren't very good,
right? Haha. So you got B plus on the quizzes

that would translate to the number 3.5 on
that four point scale. And let's say you got

to be on the final. That's too bad, but that's
3.0. So what do I say that's too bad? Well,

you probably want an eight because the final
counts for greater weight right accounts for

70% and you'd want that to be really high.
Great. Now I first wanted to show you the

non weighted average, like the normal mean,
you would make the normal mean you would make

as you just add the four to the 3.4 to the
three and then divide by three, because you

have a three in there, and you'd get 3.5,
you get a B plus in the class, right? But

let's just look down, or let's look up at
that formula. So this is the weighted average

formula. It's the sum of x times the weights,

divided by the weights. And remember what
I said sum of x y, like as an example. So

we have to, instead of just summing x, like
we did in the non weighted average, we have

to do X times W, on all of them in summit,
and you're like, what's w? Well, remember,

I told you what the homework worth 10% that's
the weight for it, right? And so, so using

percent, when we do the weighted average,
you use the decimal version. So you'll see

under the weighted average, I'm doing that
sum of X w thing by taking the four and timesing

it by point one for that 10% first, and then
see that B plus that 3.5. That gets multiplied

by point two, because that's where 20% and
then there's that B, you got on the final,

right, that gets multiplied by point seven.
So that's the sum of X w thing going. And

what do you get, you get 3.2. Now I don't
even bother to, to divide this by some of

W, because some of W is one in this case,
like if you add up point seven plus point

two plus point one, you get one. And that
often happens, you just make the weights add

up to one. But I just wanted to let you know
if for some reason you had goofy weights that

didn't add up to one, the last thing you have
to do is divide by them. So as you can see,

in the lower part of the slide, the sum of
X W is 3.2. And if we divided it by one, we

get 3.2. And now you don't get b plus in the
class, now you get like a B. And that's the

difference between the non weight and the
weighted average is the weighted average weighted

this final be extra, and then that caused
the grade, the final grade to be lower. And

that's what waiting is. Now, I just want to
say a few things. I've gone through all our

measures of central tendencies, but I wanted
to talk about how they relate to the distributions

we learned recently. So I just put up an example
of a normal distribution. And then I color

coded these lines. So see on the way, right,
there's a color coded mean. And then there's

a green median. And then there's a purple
mode. Technically, they should all be right

on top of each other. But you can see them
if I did that, so I just wished him up next

to each other. what the point is, is if you
have data with a normal distribution, all

these three things are on top of each other.
And what the magic of this is, is you don't

even need a histogram to know. So like I use
statistical software, and I'll feed in the

data, like a quantitative variable. And they'll
say, Tell me the mean, median, and mode. And

then it will, it'll tell me the mean, median
and mode. And even if I don't look at the

histogram, if it says almost the same number
for Mean, Median mode, I automatically know

it's a normal distribution. Well, that's not
the case with skewed distributions. So with

skewed distributions, the measures of central
tendency are not right on top of each other.

In fact, they're in a different order, depending
on whether we have right skewed or left skewed.

So at the top of the slide, I've got an example
of a right skewed distribution, right? Because

it's light on the right. Alright, so what's
happening here? Well, the mean, is getting

dragged around by that tail, that big tail.
So you can see that the blue mean, is on the

right side of the median. So the median is
more resistance. So it's sort of hanging out

closer to the bottom of the data. But the
the tail, that right tail is pulling the mean

up. And then the mode is the lowest one. So
if I get this print out, and I see that the

mode is the lowest the medians in the middle,
and the means the highest, I can say without

even looking at the histogram, this is probably
right skewed. Now let's look at the bottom

of the slide where we have the left skewed
distribution, you know, because it's light

on the left, and you see the same phenomenon,
but it's going the other direction, that that

tail, that's towards the low end of the data.
It's dragging the mean down now. And notice

the median is more resistant doesn't get dragged
down as much. And of course, the mode stays

at the high part of the data where there's
more data, right? So if I get the printout,

and I see that the mean is the lowest and
the medians in the middle and the most the

highest I'm like Okay, all right. have to
look at the histogram. And I know this is

left skewed. So this is basically what I wanted
to tell you about the, the distributions,

and these actual numbers and how they sort
of relate.

So in conclusion, what this lecture was mainly
about was the measures of central tendency,

right? mode, median and mean, and how to calculate
those. And, you know, I've been kind of bagging

on the mean, I'm sorry, but the mean is just
not resistance is totally not stable. And

the median is, so you want to remember these
things? Yeah, you can kind of fix things by

doing the trimmed mean, we don't really like
to do that in healthcare. Because we lose

some of our data, we find other ways of fixing
the fact that our mean, maybe kind of goofy.

But they're outside of this lecture, how we
do that. I also showed you about weighted

average, you know, just in case you have to
hand calculate your grade. I'm actually I

had a student in my class once. And this is
back when we had Blackboard. And there was

something wrong with Blackboard. So she was
really upset because she thought she was getting

a really bad grade. But she was getting a
bad grade because she didn't do a good job

of learning weighted average, because when
I showed her how to actually calculate her

grade, it turned out to be a B, I remember
she was crying. Because she did an unweighted

average, she was crying in my office. And
then I just showed her how to do the weighted

average. And she stopped crying, she was getting
a B. So just don't cry. Try the weighted average

first, okay. And then finally, I went over
distributions and measures of central tendency,

and just related to you how the distributions,
how the numbers we get from the measures of

central tendency, how we can put them on distributions
and see some information about the distribution.

All right, well, you made it through the measures
of central tendency, get ready for 3.2 measures

of variation. Hello, and welcome to chapter
3.2. It's Monica wahi. Library college lecture.

And I'm here to go over with you measures
of variation. Alright, right, here are your

learning objectives. So at the end of this
lecture, the student should be able to state

three different measures of variation using
statistics, you should also be able to explain

how to calculate variance and standard deviation,
which I'll give you a hint, those are two

of the measures. All right, you should also
be able to calculate the coefficient of variation

and explain its interpretation. And finally,
you should be able to state chebi shows theorem.

So now we're going to be concentrating on
measures of variation. And the first one,

I'm going to talk about his range. And then
I'm going to talk about variance and standard

deviation, which are two different ones, but
I'm going to talk about them together. And

you'll see why. Then we're going to go over
the coefficient of variation, which is abbreviated

CV, then we're going to talk about Chevy,
Chevy Chevy came up with a theorem, we're

gonna talk about his theorem. And then his
theorem leads us to calculate these intervals.

Remember, intervals are like, have a lower
limit and an upper limit. I'll remind you

that and when will calculate Championship
at intervals together? Alright, let's get

started. So let's think about variation. Okay.
What is variation even mean? Well, it means

how much does the data vary? So imagine I
taught two classes, which isn't too hard,

because I do teach two classes, I teach two
of the same classes, two different sections.

So imagine that I gave a quiz. And the same
mean grade was in each class. Okay. And I

said that, could we tell how internally consistent
those grades were? So for instance, let's

say that I gave a five point quiz. And the
mean, in each class was three? Do we really

know how many people got something far from
three, like, maybe in one class, people got

a lot of fives, and ones. And that's how we
got the average of three. And maybe in the

other class, everybody just got three, like,
we really can't tell from a measure of central

tendency like median, or mean, or even mode,
we can't tell how internally consistent the

data are, especially, we can't even tell that
from a mean, two different classes can have

the same mean, and a totally different kind
of variation behind the scenes. So when you're

talking about quantitative data, and you have
a whole data set, and you do the measures

of central tendency, like Mean, Median mode,
it doesn't tell the whole story, you have

to also add on the information about variation.
And these calculations that we're going to

learn here in this lecture are about ways
to express how much the data vary in the data

set. And it's just separate from central tendency.
So central tendency is just about central

tendency. And then this variation is about
variation. And you need to know both before

you can really evaluate your data set. So
we'll get started on talking about ways to

calculate these measures of variation.

So um, As I said, I'm going to go through
range. First, I'm going to talk about variance

and standard deviation. And I just want to
remind you, you know how I'm always going

on about sample statistics versus population
parameters. Well, this starts playing in in

that the formulas are slightly different than
for sample variance, the standard deviation

and population standard deviation. So we'll
go over those separate

different formulas.

Finally, we're going to talk about in the
measures of variation, we're going to talk

about the coefficient of variation or CV,
but we'll do that after these other ones.

Okay, so we're going to start with the range,
because it's the simplest to calculate. So

here's how you do it. So you'll notice on
the right, I just made up five numbers, I

just totally made them up. I don't know what
they are. Okay, I just did that for a demonstration,

because the range is the difference between
the maximum and minimum value. So literally,

it's pretty easy to calculate, you have to
first search around for the highest or the

maximum, which in this little data set, it's
so cute. It's only got five numbers. So it

was obvious that somebody ate was the highest,
right? And it's sort of obvious at 21 is the

lowest. So how you calculate the range is
you take the highest minus the lowest, and

then you get a number. And that's the range.
And sometimes my students actually take the

highest, and then they put minus and then
the lowest. And then they tell me, that's

the range. And I'm like, No, yeah, I actually
have to subtract it out. So you'll see here,

it says 78 minus 21 equals 57. So it's 57.
That's the range. Okay. So all it's telling

you is the distance between the top and the
bottom. And I'll just say that, that's not

very useful. In fact, I had a problem with
that when I was working, I worked at the army

on this army database. And I looked at the
range of ages of soldiers when they started.

And the range was h Four, three 107. Alright,
obviously, there was a problem with the data,

right? Just for some reason, there was a screwed
up record that said, somebody got him when

they were four. And there was another screwed
up record that said, somebody got in when

they were over 100, they were just screwed
up data, okay. And that caused me to have

this ridiculous range. And so the range is
not very stable or resistant, right? If we

just fixed that, you know, record that said
somebody was four when they got in the army,

then we might have a normal range, you know,
like little more like a minimum, we might

see 18, or 17, or 19, or something. But, as
you can see, on the right side of the slide,

I just picked out that the minimum and the
maximum, we could just change arbitrarily

change those numbers. And suddenly, we'd have
something totally different from 57. So as

you can see, even though this range is a measure
of variation, it's not stable and resistant.

And it actually kind of doesn't tell you much.
If I say we've got a range of 57, you don't

know if the minimum is like zero, or like
negative, or like 105, you know, you really

don't know where that ranges in. So it's not
very useful. But it's a place to start, because

that's our first measure of variation. Now
we're going to get into what we really use

in statistics a lot, you'll sometimes see
in articles where they state with the ranges,

they usually don't state the actual number
I tell you to calculate, they actually state

the minimum and the maximum. And sometimes
that's interesting. But variance and standard

deviation. That's what we really live on in
statistics for measures of variation. And

you're probably wondering why I'm talking
about them together when they're totally different

calculations. Well, it's because they're friends.
Okay? And how are they friends? Well, the

variance calculations, kind of a big formula.
And so you get through that, and then you

have the variance. And then all you have to
do to get the standard deviation is take the

square root of the variance. So that's why
they're friends is like you go through all

this trouble to get the variance. And then
the next step is just take the square root

of that, and you get the standard deviation.
So before I actually talk about those formulas,

I wanted to just set in your head, what these
words mean. Because, like, I remember, I worked

in a mental health place. And I don't know,
we didn't have enough licensed people there.

And so our leader said, Oh, I'm applying to
the state for a variance, right? Meaning that

the state would give us allow us to vary from
the rules. Well, that's what variances is

how the data vary. So you think of the spread
of the data and how well does the mean every

represent that spread? It doesn't, right.
So variance is a way of representing how the

data vary really around the meet. Now, you're
probably wondering, well, then why do you

even have standard deviation? It's the square
root of variance. But let's just think about

what the word means. You know, standard means
sort of following a standard are the same.

So it's just the amount of variation, that
standard in the data set. And you know what

the word deviation means? Like, you can say,
oh, that person is a social deviant because

they go to crimes or something. Or like this
guy with a healthy nose, he does not have

a deviated septum. But you know, some people
do have a deviated septum where it's like

crooked,

right and they have trouble like sneezing
and blowing their nose and sometimes even

breathing. Well, a standard deviation would
simply mean that everybody's deviation is

about the same. So, variance is a calculation
that says how much things vary. And so the

standard deviation, because it's just the
square root of variance, but I just want you

to imagine in your head, oh, standard deviation,
that means how much the data deviates around

the mean, because a lot of times students
get confused about the measures of central

tendency, they try to apply them to variation,
but variation is totally different thing.

So just remember what variance literally means,
and what standard deviation literally means.

And that might help you get through these
formulas and understand the interpretation.

So as I mentioned earlier, the formulas for
variance and standard deviation are different,

whether you're talking about a sample, or
a population. And, admittedly, we don't use

the population variance or population standard
deviation calculation very often, because

we don't measure the population that often.
So we tend to use the sample variance and

sample standard deviation all the time. So
I'm going to demonstrate those. But you'll

notice conceptually, they're really similar.
Like, um, you know, if you have population

parameters like Meuse, and like population
standard deviations, they tend to behave similarly

in formulas, as sample versions, it's just
that in statistics, we always want to be really

clear about what we're talking about. So we
always want to use the right symbol, so we're

hinting towards, we're analyzing a sample
versus we're analyzing a population even though

conceptually like means or a mean, right?
But you want to represent which mean you're

talking about one, that's a parameter, or
one, that's a statistic, whenever you write

out the formula, so I'm just being picky about
that. And then there's also two other things

you want to know. Um, there's two different
ways of actually doing each of these formulas.

You know how like an algebra, you can have
a big equation, and you can express it more

than one way. So that's all they do is they
put a formula in one way called the defining

formula. And then they put the formula, same
formula, but rearranged by algebra into the

computational formula. Now, I always think
that's kind of funny that they call the computation,

right? I mean, both the formulas give you
the same results, it's just plugging in numbers

and getting out the answer. And the answer
is gonna be same, whether you use the defining

formula, or the computational formula. But
what I think is so funny is they call it the

computational formula, but I cannot compute
it. Like I always get confused when I use

it. So I pretty much ignore the computational
formula in my entire life. And I just teach

the defining formula. And I find my students
always remember the defining formula, they

always can get through it. Although people
who are into the computational formula, they

tell me that I'm doing things the hard way,
I'm going the long way around. But you know,

what just goes a long way around, it helps
you not get confused, and helps you convince

yourself you actually got the right answer.
So let's just do the defining formula. All

right. So let's look at the defining formula,
you can look it up, you can look up the computational

formula, but this is the defining formula.
So let's just get get our minds wrapped around

that. Remember, I told you that variance is
great, because you calculate that, and then

you just take the square root of that, and
you get the standard deviation. So as you

can see on the left side of the slide, we
abbreviate the sample variance by just saying

s, which is the standard deviation to the
second. I know that sounds ridiculous, right?

Like why don't we have a special thing just
for the variance? Why do we just say it's

so the second and then say sample standard
deviation is just as as well actually, to

be honest with you people use different notation.
I'm just using this because it matches the

textbook we're using. But people will often
say var for variance. And so in other textbooks,

they'll do that, and then statistical software,
but they'll also say s to the second like

this, and it's maybe a good way of you remembering
that the standard deviation is just the square

root of the variance, right? So if you ever
see s to the second, remember, S is the sample

standard deviation, and s The second is the
sample variance. And I'll show you the population

one in a minute. But if you see those, that's
what they're talking. Okay. Now, let's look

upstairs at the top formula. See this thing
on the top? It's really kind of scary, but

we're going to work through this and you're
not going to be scared of it. Okay.

I know you know that there's a little some
sign there that capital sigma so you know,

something's gonna They get summed up. But
that looks kind of scary that x minus x bar

to the second thing will handle that, okay.
But n minus one on the bottom, that's not

so scary, okay. And we'll handle that one
too. And then you'll just notice, all I did

for the bottom part is I just put this huge
square root sign over that whole thing. So

that's the only difference between the upstairs
and the downstairs. And then I also wanted

to show you a picture of a calculator, because
a lot of times, if you haven't really done

math or statistics for a while, you forget
the whole concept of square root. And I'll

just remind you, whenever there's a square
root of something, it just means that if you

times it by itself, you'll get that number.
So remember, like 25, the square root of 25,

if you put 25 in your calculator, and you
hit that square root thing, you'll get five,

right, because five times five is 25. However,
if you put in 24, you're gonna get something

with decimals, right? But whatever it is,
you get, if you times it by itself, you'll

get 24. So I just want to remind you of that,
because sometimes people forget that if they

haven't been doing statistics or math for
a while, or they haven't used the calculator

for a while. All right, I told you, I talked
to you about this numerator, right that the

top is the numerator in a fraction, and the
bottom is the denominator. So I'm going to

talk to you about this numerator. So the sum
of X minus X bar squared, you know, that's

how I would say it, this is actually called
this little piece of the formula is called

the sum of squares. And so when From now on,
when I say sum of squares, I literally mean

the top half of this equation. So what you
do when you do the defining formula, is you

just kind of relax and say, the first thing
I'm going to do is figure out the sum of squares,

I'm going to figure out the top part. And
then I'm going to just write that down, and

then later, I'm gonna come back to this formula
and enter it. So this next part is, how do

we figure out that top part of the equation?
How do we get the sum of squares, and I'll

show you. Okay, so let's just look at the
slide, I'm on the left, there's this blank

table. And that's usually what I do first
is I make this blank table. And you don't

want to say column one, column two, column
three, I just put that there. So I could talk

about the columns. And then you know, I was
talking about, but usually, what I put is

I put x in the first column, and they put
x minus x bar I wrote out minus, but you can

just use a dash. And then I put in parentheses
in the third column, x minus x bar to the

second, like that. Remember, when you have
parentheses, you have to do what's inside

the parentheses first. So this means you literally
have to do X minus X bar before you to the

second it or square it. And I'm just walking
you through this to get you ready for what

we're going to do with this tape. On the right,
so this slide, I'm just reminding you that

the sum of x minus x squared to the second,
in other words, the sum of whatever is going

to be in the column three. That's another
way of saying the sum of squares. Okay. So

an easy way to explain this, what the squares
are, is to just show you how to calculate

it.

So I just

pulled out some data set, imagine a sample
of six patients presented to Central lab.

So this happens to me when I go to my doctor,
sometimes she'll say, you know, it's time

to do a lab panel for you. So she gives me
this slip of paper, and I go downstairs to

the central lab, and I give them the slip
of paper, and they say, Okay, sit down, and

then we'll call you up, and we'll draw your
blood or whatever. So we're imagining six

people did that. And then they got up to have
their blood drawn. We asked them, How long

did you wait? Okay. And I'm in the central
lab where I literally do wait two minutes,

that's a really good

lap. But

sometimes it's really busy if I go like during
lunch, and I'll wait something like 10 minutes.

So here are six patients. One of them waited
two minutes, a couple of them waited three

minutes, probably the other three came in
during lunch because they waited eight minutes,

10 minutes and 10 minutes. Okay, so that's
our data, that it's a little tiny data set,

but I just wanted to use something small to
show you how to calculate the variance, and

then the standard deviation with just this
little data set. Okay. So what's the first

step? After making the table you have to make
the blank table for us is you fill in the

first column, which is called x. So what is
x? Actually, each of these patients waiting

time is an X. Remember sum of x, if we said
sum of x, we would mean add all these x's

together, right? So So that's all I did, I
just put each x in the column, you'll see

2338 1010. It's just like identical to these
x's. And then I put at the bottom, I put that

little fancy sum of X and said 36. Okay, and
so that's just the first thing you do. Just

put them all in and do the sum of X. All right.
Now the next step is don't look at the left

side of the slide yet, look at the right side.
Before you go and fill in column two, you

have to do X bar. In other words, You have
to figure out the mean. Now, you can kind

of cheat because you just figure it out some
of x. And if you remember the formula, the

mean, or the x bar of the sample is the sum
of x divided by n. And remember, I told you

at six patients, so you just take 36 divided
by six, and you get six. Now you just hold

that number,

you hold that.

So between column one and column two, you
got to calculate x bar, and you hold, right.

And then while you're holding that, you keep
it off to the side, you realize that this

is how we're going to fill in column two is
what x minus x bar means is the x bar is just

six. But we have to go through each x and
minus x bar from, it's helpful to order the

x's before you do this, like notice, I put
them in order 2338 1010, it's a good idea

to just do that, because it helps your brain
think whether or not you're doing the right

thing. So let's start with the two. So we
do two minus six, which is the x bar. Now

you can look at column two, two minus six
equals negative four, I hate negative numbers,

but you just have to deal with them sometimes.
Okay, so it's negative four, so you just deal

with that, then you go to the next slide,
and it's three minus six, which is negative

three. So we're still on the water here with
the negatives, but you'll notice that the

next 1x is three, so you can kind of copy
what you just did. So you're getting negative

three. So what you're actually technically
filling in this column, I showed you the equation,

but you're putting negative four in the first
one, negative three in the second one, negative

three in the third one. And then now finally,
the fourth x is eight. So eight minus six,

we got above water, now we're in two, right.
And then we have 10 minus six was 410 minus

six, which is war. And when you order them
like that, that's often what happens. In fact,

that's always what happens is you end up with
a bunch of negative ones at the beginning

and a bunch of positive one later, that's
just totally normal. Don't worry about that.

But you got to be careful, you got to make
sure you make the right meet. I've had people

on tests actually screw up this mean. So you
can just imagine when a train wreck happens

after that is you do not get anything right
after that. So make sure your means right.

And then make sure you subtract it from every
single x and put the right answer in column

two. That's the next step. All right. Okay,
so we're done with that step, what do we do

next? Now, we just take whatever we got in
column two in square. So we have the first

one was negative four. So we take remember,
square is just the the number time itself.

So if you don't like to use x to the second
button on your calculator, you can just do

negative four times negative four, same thing.
And so you'll notice we do negative four times

negative four, we get 16. Now, it's pretty
easy. negative three times negative three

is not, you know, two times two is four. But
I what I want you to really look at is the

10s. Notice that they get a 16 two, just like
the two did. And that's

the trick here. Remember, I said I hate negative
numbers? Well, a lot of statisticians feel

the same way I do.

And so they often fix it by squaring the number
because it's a racist, the negative. Just

remember, negative times negative is positive,
and positive times positive is also positive.

That's a little trick, you know, when it comes
to multiplying. And so when we do that, we

are squaring each one of column two. And they're
called squares, right? So we've got 16 994

1616. These each are squares. So what do you
think we do? We add up that entire column,

and we get the sum of squares. So look at
that, we add up that entire column, and we

get that super complicated looking thing at
the bottom, which is the numerator for our

variance equation, right? Like this wasn't
really that hard. Was it? Okay, so we sum

that up. And as it turns out, we get the number
70. So 70 is our sum of squares. All right.

All right. Now we're back at the sample variance
formula. And I'm so excited because look at

the top of the formula. We answered. It's
it's 70. Okay, so we got that 70. But we still

have to deal with the bottom of the formula.
Remember, n was six, right? We had six patients,

and the bottom of the formula is n minus one.
So the bottom of the formula is going to be

five, right? So let's fill this in. I was
kind of running out of room, so I just filled

it in upstairs. So you see that 70 divided
by five suddenly this looks super easy, right?

So 70 divided by five is 14. Okay? That's
the variance. totally easy, right? Once you

make that, I mean, it's not it's tedious,
right? You have to make that whole table and

add things up and stuff. But here, it's not
really that hard. Now, Guess how we're gonna

make the standard deviation you've probably
guessed it, we're just going to take a square

root of 14. So remember that button on your
calculator, you could put in 14, hit that

button, and you get 3.74 and a bunch of other
stuff, but I just chopped it off at 3.74.

So that is your sample standard deviation.
Now I promised you I would talk about the

population formulas for standard deviation
and variance, as well as the sample ones.

And I told you, they wouldn't really be conceptually
much different. As you can see on the left

side of the slide, sample variances expressed,
I made things red, so you can see what the

differences were sample variances s to the
second, but population variances as other

Greek letter. Remember, I told you that that
other sum was capital sigma, like, you know,

Greek is like English, in the sense they have
capital and lowercase letters? Well, that

thing that I always think it looks like a
jelly roll, but the Jelly Roll looking thing

is actually lowercase sigma. So that I'm never
going to say lowercase sigma, except for now,

I'm going to say population variance and population
standard deviation. So you'll see at the bottom

of the slide, the lowercase sigma alone is
the population standard deviation. And then

the lowercase sigma to the second is the variance.
So just remember, if you see that Jelly Roll

thing, we're talking about a population version
of the standard deviation or variance in that

the sample. Also, you already know about mu
versus x bar, right, so we have x bar on the

left. And that's the sample mean, and mu on
the right, which is population mean. And you

also already know about n, which is the number
in your sample. And this is where there's

a big difference actually, in the sample,
you have to do n minus one on the bottom,

and in the population, you just do, and capital
N that whole population. And if you think

about it, it makes kind of sense, because
populations are huge, so won't even matter

if you like subtracted one. Whereas, you know,
samples are small. So you sometimes have to,

you know, adjust or something, so you have
to minus one, but you wouldn't even matter

like people make a mistake and accidentally
minus one from the population one, they don't

get much of a different answer. And so that's
why I'm concentrating on the sample once,

that's what we normally do. But I wanted to
give a shout out Just so you know, if you

ever see the arm formulas on the right side
of the slide, you know their population level

formulas. Alright, now we're gonna move on,
we made it through range, variance and standard

deviation. So now we're gonna move on to talk
about the coefficient of variation. And this

is used a lot for comparisons for comparing
between two different labs often.

I say that because my friends are pathologist,
in the first time I actually use this in medicine,

as we were comparing lab values on the same
assay from two different labs, I just wanted

to explain to you this might be the first
time you've heard the word coefficient. And

that gets a little confusing for people in
statistics who are new, because the word coefficient

is actually just kind of a generic term for
certain kinds of numbers. So you'll hear somebody

say, coefficient of variation. And you'll
say, you'll hear somebody say coefficient

of something else, or coefficient of something
else. And just a word coefficient. Most people

haven't even heard it. It just means a certain
kind of number. It's just somebody says, oh,

the coefficient is not good, or it's high,
or whatever, you need to ask them, What coefficient

are you talking about, right. So in other
words, coefficient doesn't mean a specific

thing. It just means a number that comes out
of statistics. And so you have to know which

coefficient they're talking about. So this
is the first time maybe you've heard the word

coefficient. And I'm going to talk for the
first time then, to you if you've never heard

coefficient before, about a specific coefficient
called the coefficient of variation. Now,

you'll, as we go through this textbook, there's
other coefficients on it. So please remember

this one is coefficient of variation, right?
And a way to remember it is a CV for short.

And so other coefficients have different abbreviations,
but the coefficient of variation is CV. So

I put on the right side of the slide the the
formulas, and nobody seems to have any trouble

doing the formula, right, because once you
calculate the standard deviation, the sample

standard deviation of the population one,
as you can see in the formulas, and once you

calculate x bar, which is a mean for the sample,
it's pretty easy to do the division, and then

they like it when you do it in percent. And
you'll notice that about statistics is certain

things they prefer as proportions. And certain
things they prefer as percents. It's just

like, I don't know, it's just like our culture
in a way and so coefficient a very is always

expressed as a percent. So you have to times
that by 100. And then put a percent sign after

it. But really, that's pretty easy to do you
take the standard deviation, you'll see I

did it for our patients 3.74. It took us all
that work to get there, right? Remember square

root of 14. And then remember, our x bar was
six. So we needed that remember earlier for

that column, too. So I just dumpster dive
dumpster dove, those numbers, and then did

this calculation out and I got 62%. And so
students generally don't have trouble getting

that number. But what the problem is, is like,
what is the number even mean? Right? Like,

what does it mean, if you divide the standard
deviation by the x bar and times by 100%?

And like, how do you interpret that percent?
So the easiest way to talk about it is to

actually compare it with something. Because
one thing you'll also notice in statistics

is if you make ratios of things, they don't
have any units. So if I take your blood pressure,

like your systolic blood pressure, and I say
it's whatever, 130 mmHg. If I divide that

by your diastolic blood pressure, or even
by some lab value, or your temperature, or

whatever, your IQ, suddenly I get a ratio,
and that doesn't have units, right, it doesn't

have mmHg, or anything like that. And if I
do that to a bunch of people, all of those

ratios don't have any units.

And so they technically could be compared
to each other. So you'll see that that's a

strategy in statistics is they'll make ratios
of things and say all those don't have any

units. So it's, you know, sort of lacking
in that way. But the power is you can compare

these ratios. So, I decided to just pull out
other patients, I just made up other patients,

right. I pretended we went back to the lab,
the next day, and we gathered some data. And

we gather some data, and we came up with I
just made this up an x bar of eight, and a

standard deviation of four. It's a little
close to what we had before, right? Like x

bar six insanity, Visa 3.74. But anyway, in
this next sample patients, the S four divided

by the x bar of eight times 100 equal to 50%,
and not 62%, like the other one did. So how

do you interpret that? Well, the CV is a measure
of the spread of the data relative to the

average of the data. So in the first sample,
the standard deviation is only 50% of the

mean. But in the second sample, the standard
deviation is 62%.

of the mean.

So what I would say is that the second sample,
the red one with the 62%, has more standard

deviation, compared to the mean. And so that
means it's less stable, right? It's got more

variance compared to its mean, and it's more
standard standard deviation compared to its

mean. So it's less stable. So it moves around
a lot. So if you said to me, if these were

actually two different labs, I would say,
you know, I prefer the first lab, the purple

lab, because it's more predictable. I know,
it's gonna be like less variation, because

it's 50%. And the 62% means that that's less
predictable. It's a little hard to see in

this example. But what happens is, if you
have two different labs, and you're looking

at this, like maybe you split a blood sample
or a bunch of blood samples and send half

to one lab and half to the other, what you're
supposed to get the same mean and the same

standard deviation, right? They're the same
blood,

you just want it.

But sometimes you don't sometimes you get
something like this, in which case, if you're

comparing labs, you would go with the purple
lab and not the red lab because they produce

a more predictable result. So CV is a little
hard to interpret. But it's easy to calculate.

So that's one awesome thing about now, we're
gonna move on to chubby chef and his theorem.

So chubby chef figured something out a long
time ago. And this is how he started thinking

about it. He first started thinking, well,
let's say you have an x bar and an S, like

we just did with the CV. He noticed something
else about it, he didn't notice the CV, he

noticed that you can create a lower and upper
limit by subtracting the ass and adding the

s to the x bar. So remember back when we were
making frequency tables, and I said, Well,

we need to make class limits, we need to make
a lower class limit and an upper class limit.

Well, we use those terminology a lot like
lower limits and upper limits. Well, Chevy

show was like wait a second, I got an idea.
Let's say I take a mean. And I you know, this

will force the mean to be in the middle of
this. I can subtract one standard deviation

from it, and I'll get some sort of lower limit
and I'll add a standard deviation to that

mean and get some Sort of upper limit. And
of course, let's pretend the standard deviation

was one, like you'd subtract one to that one.
And so this would be like totally symmetrically

in the middle, right, the x bar would be in
the middle, and then it'd be surrounded equally

by these two standard deviations. And I'm
just saying standard deviation generically,

because you could do this with a mu, and the
population standard deviation, two, you can

do the population work. So he just sort of,
like figured out, that's a thing that can

happen, you can add and subtract a standard
deviation from the mean. And you can get these

limits. And so example, let's say I have a
mu. So I'm gonna pretend I have a population

a mu of 100. I don't know what I measured,
but I got 100 and a population standard deviation

of five. So Chevy, I was thinking, you know
what I could do, I could take that 100 and

subtract that five from it, and I get 95,
I could take that 100 and add five to it,

I get 105. And so we just started like working
with this concept, like I could subtract and

add like a standard deviation. And then he
thought, Wait a second, I could even do this

with two standard deviations, right? So I
could take like, if it was five, I could take

that times two, that's 10. And so I could
do 100, subtract 10, and I get 90 for the

lower limit, and 100 and add 10. And I get
110 for the upper limit. And so I can make

this this range or this interval, right? from
the lower limit to the upper limit, we call

it an interval, right. And so he just sort
of conceptually realized that if he used some

rules along with this, there might be some
useful interpretation of these limits, right,

there might be some way that uses limits to
mean something. So we're going to look at

how he figured out to be able to use, you
know, one standard deviation on either side

of the mean, or two, or three, or four multiples
of these standard deviations on either side

of the mean, to actually come up with some
lower and upper limits, that meant something.

So he realized that what these low lower and
upper limits would mean is that at least some

percent of the data would be between these
limits. So in other words, some percent of

the of the axes would be between the lower
and the upper limit. But that percent would

depend on how many standard deviations you're
going out, right? Like is it one is a two

is a three, the, the more you go out, obviously,
the more percent of your data are covered

by the limits, because they're just huge,
like, get it. So the interval so big, and

almost covers the whole thing. So you would
expect that percentage go up, as the number

of standard deviations you use goes up. So
so he was working on this out, and he came

up with this formula, right. And he also,
he was figuring out, he wanted this to work

for all distributions, like normal, but also
skewed. And also like uniform and by modal.

So this was the formula he came up with. Now,
in this formula, see at the bottom, k stands

for the number of standard deviations, or
the number of population standard deviations

that he's going to use, right? So let's pretend
that he made KB to like two standard deviations,

right? Then you'd see this, it says one minus
one divided by k to the second, which would

be to the second, so that would be to the
second is what four. So one divided by four

is point two, five. And so one minus point
two, five is like point seven, five, well,

you make that a percent at 75%. So

he's like, okay, that's what I'm going to
say. If you go out two standard deviations

up or down, and you make those upper and lower
limits, at least 75% of the data of the axes

are going to be there, at least, there might
be more, but it'll be at least that. So he

did this he used to, and they use three, and
he used four.

So

two standard deviations, either way, three
standard deviations either way, or four standard

deviations either way. Now, students in my
class often think that they have to memorize

this one minus one over K to the second, you
don't memorize. This was just a story of how

Chevy chef did this proof. So you can memorize
it for fun, but nobody memorizes it. I mean,

you know, Chevy chef did the work. I'm just
showing you the proof, right? So he figured

this all out. So as you can see how he like
you can do this with two, three and four,

you'll get the same answers Chevy chef does.
So it's kind of a waste of time, but you can

do it just for fun. So he did the two one,
I showed you that on the top. I even talked

you through it. So you've plugged two into
the equation, you'll get 75%. So in that thing

I was just talking about like imagine I had
100, right? And that was my x bar and my standard

deviation was five, right? And then two times
that is 10. So I go well my lower limit then

would be 90 in my upper limit. That would
be 110. And I would be able to confidently

say at least 75% of my x's are between 90
and 110. So if I'd measured maybe 100 people,

right, I'd say at least 75 of them are going
to be between these limits. In fact, it could

be 80, could be more, but at least 75. So
then Remember, I told you to predict that

as we made this number bigger, you know, we
go out more standard deviations, we're going

to cover more of the data, right? So we needed
three, it didn't come out as even, it came

out in 88.9% of the data. So almost 89% will
be covered if you go out three, and at least

almost 88.9%. And if you go out four standard
deviations, it's at least 93.8%. Right? And

just to remind you, you know, when you have
upper and lower limits, you have an interval,

right? That's just we just call it that. But
this particular interval, if you get it this

way, it's Chevy service interval, because
everybody's so happy did all this work, right?

Because I wouldn't have figured it out. So
I just wanted to demonstrate an example of

championships interval, because then you can
know how to interpret them or why anybody

does them. Okay, so remember our patient sample,
they're in the waiting room at the lab, right?

So they waited on average, six minutes, and
then the standard deviation of them waiting

was 3.74. Right? Now, when I gave you this
demonstration of how to calculate the standard

deviation, I use this patient sample, I did
that I only had a few patients in the sample

on purpose, because otherwise your table that
we made with the defining formula would be

huge, and I never finished this video. So
what I'm gonna ask you to do is pretend that

instead, we had 100 patients in there, right?
Instead, I measured 100, and I got my x bar,

my 3.75 standard deviations, okay, so if we
measured 100 patients, and we got that, I

just want to, I put this chubby shove rules
in that table. So if we go out two standard

deviations from the mean, from the x bar,
either side, whatever limits we get whatever

interval we get, we know at least because
I made it, so we say you know, studied 100

patients. So by law, we're at least 75 of
those patients will be between those lower

and upper limits, if we follow championship
syrup. And if I do go out three standard deviations,

at least 88.9 patients will be in there. Okay,
I know that doesn't make any sense, like 88.9

patients Saudia point nine of a patient. But
what they're saying is, I guess it would be

89. All right, yeah, 89% of the patients or
in other words, 89 patients, at least would

be in that interval. And of course, if I went
out for at least, I wouldn't have to say 94.8

of a patient, but at least 94 patients would
fit in that interval. And if you're thinking

about if we only start with 100 patients,
that's almost all of them. So the for one

isn't so useful, right? So you'll see me on
the left side of the slide calculating the

intervals, right? So let's start with the
first one. The first one is two standard deviations

on either side of the mean. So the chubby
chef interval we get is negative 1.48 to 13

4.48. And you probably notice you can't wait
negative time. So already, this is kind of

weird, right? But what this is saying is of
our 100 patients, at least 75 of them because

this is 75% championship interval, weighted
between negative 1.48 minutes, so that might

as well rounded to zero between zero minutes,
and 13 4.48 limp minutes, right. And so at

least 75% of them are, I fell in that range.
Now 13.48 minutes is kind of long. So we would

be happy, I guess is 75% of them fell in that
range, because then that means

that they were probably not waiting that long.
But if you go out, then you widen this interval

like 88.9. If you do that, then you say at
least well rounded to 89 89% of the patients

waited between negative five point to two
minutes, which is you might as well make zero

and 17.22 minutes. So as you see, if we widen
the interval, we're going to get some later

waiters in there. And so then we'll say, Well,
at least 89% were between there, but at least

90 89% were between there and that means it
wasn't bigger, right. And then again, we go

out one more, we get 93.8%. So let's just
round it to 94. So at least 94% of the patients

or if we have 100 patients, at least 94 of
them waited between negative 8.96 minutes,

which again is nonsensical, up to 20.96. But
then we're starting to get where we'll have

almost all the patients with Somewhere between
zero and 20 minutes, we really don't know

how long they waited. So this is just kind
of to show you what happens when you line

that interval, you you maybe have less certainty
about what individuals happen, be sort of

a better idea of what the range is. So again,
I just put this at the bottom. If we had 100

patients, this is how you would interpret
it, at least somebody five would have waited

between the lower and upper limit for the
75% championship interval. And then at least

80.9 patients I know nonsensical. And then
the 93.8. So you see that interpretation lower

part of the slide. So this is a really difficult
concept for a lot of students. And so I'll

just give you this take home message. First
of all, Chevy shove interval works for any

distribution, normal skewed whatever. Reason
why that's part of the take home messages

later, we're going to learn about intervals
that only work with normal distributions.

Okay? So this one is loosey goosey. It works
with all distributions. So that's one of the

take home messages for chubby sets interval.
Also, Chevy says interval tell you that at

least a certain percent of the data are in
the interval. Later, we're going to learn

about intervals where exactly a certain amount
of data are in that interval. And so Chevy

shop again, a little loosey goosey, right,
he says at least. Next, championship intervals

are sometimes nonsensical, as we just talked
about. Negative time doesn't work, right.

Sometimes you'll have very high limits, especially
with a four. And so ultimately, they're not

very useful. And they're not used in health
care. I literally had never heard of Chevy

shows interval until I started teaching this
class. So what is the purpose of teaching

you Chevy says interval. The purpose of teaching
this is to point out in statistics, we often

use the s or the population standard deviation,
you know, just standard deviation. And we

add or subtract, we'll add and subtract it
from the mean, is a good way of making lower

and upper limits that have special significance.
That's really the main take home message is

that you'll see this pattern as we go through
this class, where we get a mean either populations

or sample, and we have x bar, you know, x
bar or population mean. And then we have a

standard deviation, right either from sample
a population. And then we take either one

standard deviation, we added subtracted or
two, or multiples. And those intervals then

have certain significance. I only taught you
in this one about Chevy chef, what you learn

about other intervals later that are made
similarly. So in conclusion, what did we learn,

we learned how to calculate the range, we
learned how to calculate the variance and

standard deviation. We learned about how to
calculate the coefficient of variation, how

to interpret it. And we talked about the difference
in the formulas from sample versus population.

And we learned about Chevy Chevy and his theorem,
how he figured it out, and how we calculate

this intervals and how you interpret them.
Now I just thought I'd show you this picture

of Chevy chef here. He's a Russian guy. Well,
the stamp was from the USSR, for the Iron

Curtain fell. But I just thought I'd show
it to you. So you knew who figured all this

out? Good job, you've made it through the
measures of variation. And now you're ready

to do what the quiz, the homework, whatever,
right? You're totally knowledgeable.

Good job.

Well, I'm back. And so are you. Welcome to
Chapter 3.3 percentiles and box and whisker

plots. It's Monica wahi. Library college lecturer.
And this is what we're going to talk about.

And this is what you're going to learn. At
the end of this lecture, the students should

be able to explain what a percentile means,
describe what the interquartile range is,

and how to calculate it. Explain the steps
to making a box and whisker plot, and also

state how a box and whisker plot helps a person
evaluate the distribution of the data. So

let's get started. You know, whenever we talk
about a box and whisker plot, I think of some

cute little animal with all those whiskers.

I'll explain what the whiskers really are,
I mean, not on the animal, but on the box

and whisker plot later. So what are we going
to go over, we're going to go over percentiles,

and we're going to explain what those are.
Then we're going to talk about core tiles

sounds a little slimmer, it's got the tiles
and it will you'll you'll understand why they're

similar. Then we're going to compute core
tiles. And then finally, we're going to do

the box and whisker plot. All right. So let's
go. So percentiles, we're going to have a

flashback, okay. You're not going to like
this little part because it's going to remind

you of standardized tests. So maybe not all
of you have been subjected to this, but most

of us have if you gone to high school. In
the US, you probably got to deal with these

standardized tests. So just remember, we're
only talking about quantitative data. All

right. So if you take a standardized test
or a non standardized test, you usually get

points. And points are numerical. So that's
quantitative

data.

So I remember I used to take the standardized
tests, and I'd be, you know, showing my friends

what I got, right, because they'd send you
that thing in the mail. Now, I learned pretty

early on, that it mattered who all was in
the pool of people maybe taking a test with

you, right. So if you're taking the test with
a lot of stupid people, it's easier to get

a higher percentile, because what percentile
means is it for example, if you test at the

77th percentile, it means you did better than
77% of people taking the test. And a lot of

those standardized tests, they didn't care
how many points you got, what they cared about

is what percentile you were at. So different
batches of people would have different scores.

And if you got a lot of lucky, got a lot of
stupid people, then your score would be higher

than there. So it didn't really matter what
your absolute score was, it just mattered

what your percentile was. So just to sort
of remind you, if somebody had come up to

me in high school and said, I got 77 percentile,
what I'd say is okay, if only 100 people had

taken the test, you'd have done better than
Sunday, seven of them. Of course, we were

all Brady, Brady, you know, I was always in
like the 95th, or the 97th, or the 98th. And

it happened so often, I wondered if it was
really true. But what I realized is, is that

there were so many people in the pool, because
you know, I was in public high school in Minnesota,

well, they were pulling together all the public
high schools in Minnesota, ninth grade, you

know, as pulled with them in 10th, grade or
whatever. And when you're taking like nursing

examinations, sometimes they'll do that they'll
put you on a percentile. So I try to tell

people, you know, strategize, try to take
in when only stupid people are taking, which

of course, makes no sense. How can you tell
when stupid people are taking it, right? You

don't even know who's taking it. But really,
that's that's what a percentile is, it's the

percentage of people that you did better than
if you're at the 77th percentile, then you

did better than 77%. Okay, so here's just
some rules about percentiles. First of all,

you know, I gave the example of the 77th percentile,
well, the rule is you have to have one between

one and 99. Like, you can't have the negative
second percentile, or the 100, and fifth percentile.

So that's the first, then whatever number
you pick, like I was saying, that percent

of the values would fall below that number.
And 100 minus that number, have the values

fall above that number. So like, in my, well,
here, we'll give an example. 20, people take

a test, just 20, right, let's say there's
a maximum score of five on the test. The 25th

percentile means that 25% of the scores will
fall below whatever score that is, and 75%

will fall above that score. So let's say it's
an easy test. And let's say out of my 20,

people, 12, get a four, which is almost the
total, right, and the remaining eight, get

a five, so everybody gets either a four or
five, well, then, you know, the 25th percentile,

or the score that cuts off the bottom five
tests, right, will be a four, just because

this was an easy test. And every you know,
the first 12, people got a four and then the

rest eight out of five. So even the 50th percentile,
then would technically be at a four, right?

Now, this would all come out differently if
it were a hard test, and most people got a

score below three, right? And so the percentiles
would be shifted down, I just tell you that

so you can keep in mind the difference between
the actual score and the percentile. So the

percentile just happens to mean that this
percent of people got the score lower than

whatever your score is, it doesn't actually
say what your score was, right? So that's

what you just want to remember as we're going
to percent. Okay, now we're going to talk

about core tiles, and also the interquartile
range. Remember the tile think so this relates

to percentiles. So I put a little quarter
up there. So core tiles is a specific set

of percentiles. And you'll see why I put the
little quarter up there. It's because there's

technically four core tiles, it's just that
the top quartile doesn't count because it's

like the 100% one. And remember, it can only
go up to 99, like I was just showing you.

So we calculate the first second and third
quartile. So we have the 25th percentile is

the first quartile, the 50th percentile, which
is also known as the median, which you're

already good at, right? That's known as a
second quartile. And then the third quartile

is the 75th percentile. So those are your
courthouse 25th 50th and 75th. And technically

a 100th. But we never say that, right? Because
it only goes up to 99. So you have the first

quartile at the 25th percentile, the second
quartile at the 50th percentile. The third

quartile at the 75th percentile. And these
are actually not that hard to calculate by

hand.

So here's, like how you do it sort of an overview.
So first you order the data from smallest

to largest, because remember, we have quantitative
data, so you can sort them, so you sort them

smallest to largest. And this is feeling very
immediately, right? Well guess what, that's

step two is you find the median, because the
median is also the second quartile, which

is also the 50th percentile. So already, you
have know how to do this, right? Because you

could already do step one, and two. Now, this
is the harder part, this is the new part.

Step three is where you find the median of
the lower half of the data. Right. And so

wherever you put your median, you pretend
that's the end, and you look at the smaller

values, and you find the median of those.
And that would be the first quartile or the

75th percentile. Then finally, step four,
which you probably guessed, is you find where

your median was. And then you look at the
upper half of the data between the median

and the maximum, and you make a median out
of that part of the data, and then that's

your 75th percentile. Okay, and I'll show
you an example of us doing that. But this

is an overview of the steps. Now, remember,
range before what the range was, yeah, you

remember it, that's where we had the maximum
minus the minimum, right? And I told you,

you have to actually do out the equation and
tell me what number you get. And that's the

range. Well, we have something new and improved.
In this lecture, here, we have the inter quartile

range. Okay, so you already know about quartiles,
we were just talking about them. But inter

quartile sort of means like, within, right.
So once you have the third quartile, and you

have the first quartile, you can calculate
the inter quartile range, or RQR for short.

So if you see IQ are on here, just remember,
that's interquartile range. So that's the

third quartile minus the first quarter. And
again, I'll show you an example. It's this

is just an overview. Okay, here's the example
I promised. On the right side of the slide,

you will see a sample of data I collected,
I went to HD comm that's American Hospital

directory calm, and that provides publicly
available information about American hospitals.

So I went in, and I took a random sample of
11, Massachusetts hospitals, there's a lot

more, so I took a random sample. And what
I did was I wrote down how many beds each

of those hospitals had. Because if a hospital
has several 100 beds, they're considered kind

of a big hospital. And if they have less than
100 beds, they're considered a smaller hospital.

So I wrote all those numbers down. And then
I already did step one of making our courthouse

which is to order the data from smallest to
largest. So you'll see on the right side of

the slide, my smallest hospital had only 41
beds, and my largest hospital had 364 beds

and see I put all of them in order, they're
on the right. And so we already did step one.

So let's go on to step two. So the Step two
is to find the median, and that's quartile

two, or the 50th percentile. Now, you're already
good at that, right. And so we have 11 hospitals.

So we know that the sixth one in the row is
going to be the median, you know, because

it's an odd number of hospitals that I drew.
And so the sixth one will circle it, that's

the 50th percentile or the median, so we already
got quartile two, it's, it's funny that you

have to start with quartile two, but that's

what you have to do.

Now, I just re color coded these. So you could
kind of remember what's going on as we do

the other steps. 126 is the median. That's
kind of not on anybody's side, it's not on

the lowest side, and it's not on the highest
side. The orange ones then are considered

below the median. And the blue ones are considered
above the median. And so I just color coded

them so you can keep track of what's going
on in the next slides. Okay, now we're going

to do the 25th percentile for step three.
So the goal is to find the median of the lower

half of the data. So now you see why I color
coded it is because now we're pretending just

the orange ones exist. And we are just finding
the median of that. And we're not counting

that 126, because that's already been used.
And so now we find that 90 is the 25th percentile,

how you remember that it's not the 75th, it's
not the third one is because it's the low

one, like 25 is a low number. And 75 is a
higher number. So you go to the lower part

of the data, you find the median of that,
and that's going to be your 25th percentile.

And so in our case, that's 90 then you probably
guessed it, you go to the blue ones, right

the upper half and you go get the median out
of that. And so of course ours is 254. So

that's our 75th percentile. So what we just
did is we calculated our courthouse. We have

Our 50th percentile, our 25th percentile and
our 75th percentile. So that's what I meant

by that overview slide. This is an example
of how you would do that. And of course, I

have to give a shout out to the IQ R, which
is the interquartile range. Remember, you

just learn that. So that's the 75th percentile
minus the 25th percentile. So in our case,

that's going to be 254 minus 90, which equals
164. So that is your IQ R. So if I gave you

a test, and I asked you what is the IQ or
for these data, you can't just put 254 minus

90, you actually have to work it out and put
164. So there you go. So that's our quarterly

example. So I just wanted to step back and
give you some philosophical points on what

happens with q1 and q3, depending on how many
data points you have. Okay, so remember, the

first step of this is always to put them in
order from smallest to largest. So let's pretend

I had only drawn the first six values of my
hospitals. See how I put on the slide, I put

the position of the number, which is 123456.
And I put above the example numbers. So let's

say I was going to do the median on that,
you know, what I'd have to do is I'd have

to take 90 plus 97, divided by two. But then
the next question is, what do we do for q1

and q3? Well, given that in the example of
having six values, the 90 and 97 are mushed,

together for the median, they don't get, they
can get reused, or they do get reused when

looking at the bottom and the top half of
the data. So when we went to go to do q one

in this, we would actually count that 90 in
there. In fact, q one would be 74, because

that's the median of the three numbers below
the median right below that line. And then

the Q three would actually be 121, because
we actually count the 97 in there. So in other

words, when you have like six values, and
the median is made out of mushing together

two values, like taking the average of those
two values, those two values, they get to

double dip, they get to be in the bottom,
and the bottom line gets to be in the bottom,

and the top one gets to be in the top when
calculating q1 and q3. Now, well, what if

we had seven values instead of six? Okay,
so I just expanded and pretended we had seven

hospitals. And you'll see that I have seven
positions there. Well, this was a little like

the one we did, together with the 11 values,
where the median was clearly this 97. Here,

in this case, it's 97. So that 97 does not
get reused in the bottom in the top. So you'll

notice that q one is the middle number of
the three bottom ones, and Q three is the

middle number, the top three ones. And so
that's what happens when you have seven values.

And it's also happens when you have 11 values,
like I demonstrated with those hospitals.

But it's not super predictable. Because what
if you had eight values, we suddenly see it

gets a little complicated. So how would we
do this? Well see the first four are between

41 and 97, top four between 121 155. Well,
to make our median, we'd have to take the

mean of 97, and 121. But remember, they don't
get used up the 97 then gets to double dip

and be part of the calculation for q1, and
121 gets a double dip and Part B part of the

calculation for q3. But even even with this
double dipping, if you go down, you'll see

that there are four then numbers to contend
with, for q1. So of course, to get q1, you

actually have to mush together or take an
average of 74 and 90. And if you go up the

upper part of the data, in order to get q
three, you're going to have to make an average

of 126 and 142 are the ones in position six
in position seven. So if you're unlucky enough

to get like eight values, then you realize
you're going to have to make your median by

making an average of two numbers, your q1
of making an average of two numbers and your

q3 like that. So it's not super predictable
what's going to happen. You just have to pay

a lot of attention. Just remember if your
median is made out of two numbers average,

those numbers get to double dip in the downstairs
and the upstairs of calculating q1 and q3.

If instead your median is just one number,
like because you have an odd number of values,

then that guy has to just stay there and does
not double dip in q1 and q3 calculations.

So we can just see another example of this.
So this is nine values right? Now remember,

when I had 11 values, it was like having seven
values. I had this median and it was really

clear like we have here but even Um, the medians
of the top of the top of the data and the

bottom of the day, they were just, you know,
it was an odd number. And so it was easy to

figure that out. Well, you see here, in this
case, our median is the fifth value, and that's

121. So 121, does not double dip anywhere,
right? So we go to calculate q one, we only

have four values, because we're not counting
the 121. And then we're stuck with taking

an average of the second and third value to
get q one. And then same thing upstairs here,

between, you know, 142, and 155. You know,
those are the two middle numbers of our four

numbers at the top. And then we have to take
an average of those to get q3. So I guess

this is just my long way of saying you got
to be really careful what you're doing. First,

make sure you've gotten the median, then figure
out if that median is this kind of a median

where it's just you're circling, or it's a
medium that came out of an average, because

if it's a medium that came out of an average,
just know that those numbers are going to

double dip in q1 and q3. And if it's a medium
that was because you had an odd number of

data, it was just like in the middle, that
one doesn't get to double dip. Okay, enough

double dipping, I'm getting hungry. When I
go to that roller coaster, I'm going to get

a double dip ice cream cone. Okay, we're gonna
move on to box and whisker plot, which is

kind of like your percentiles getting graphed,
right. So let's go back to our ingredients,

we already created our box plot ingredients.
In fact, that's why I trickily went through

those portals first, because now we've created
our ingredients to make a box plot. So I just

sort of summarize what we have on the left
slot, side of the slide, say that 50 times,

hospital beds was what we were counting, the
smallest Regional Hospital had only 41 beds.

q1 was 96. a little easier. I put it in an
order cure, one was 90, median q2 was 126.

You know what I mean? I mean, cuartel, right,
like by these cues, then q3 is 254. And then

the maximum was 364. Okay, so let's make a
boxplot. And then you remember what the data

looks like on the right side of the slide.
Okay, well, now I'm going to walk you through

how you would make this box plot. So first,
you draw this thing? Well, how do you know

what to draw? Well, I usually just draw a
line and a vertical line, and then put a zero

at the bottom, and then I cheat, I go look
at the maximum go, Oh, I wonder where that

is. And see our maximum was like 364. So I
just made 400. At the top, if our maximum

had been something like, you know, I think
Massachusetts General Hospital has something

like 600 or 800 beds. If we had gotten that
one in there, and that was our maximum, I

would maybe go up to 900, you know, whatever
is a little bit above the maximum, that's

what I put at the top. So this was 364. So
I put 400, then what I did was I divided it

in half, like I see where the 200 is, I just
kind of threw that in there. And then I divided

between the 200 and the 400, a half and put
the 300. And so you can just kind of eyeball

this and draw it out that way if you want.
Okay, so I got this thing set up. And then

here we go, we're going to do the first thing.

Okay, here's the first thing we're going to
draw in q1 or quarter one. So on the left

side of the slide, you'll see a circle that's
90. On the right side of the slide, I made

this horizontal line. Now how Why do you make
that line? Well, look at how its proportion

to that that upward and down graph thing I
made, you know, with the numbers, you probably

don't want to too wide, but you don't want
to too skinny. This is just about right, like

Goldilocks just right. Okay, so you just make
this horizontal line at q1. So that's the

first. Now you make a copy of that same line
parallel, and you make it at q3. So if you

look at that, if you're I hope you're not
lost, if you look at that, you know, 100 200

300 400, you know, q1 is 90, so it's about
10, under 100. So that's how I knew where

to position that lower one. And then 254,
that's about, you know, a little bit higher

than halfway between 203 100. So that's where
I roughly knew how to position this one. It's

not perfect. If you do it in statistical software,
they put it out and it's perfect. But for

demonstration purposes, that's

what I'm doing.

Okay, so now what we've done is we put in
q1 and q3 and we put these horizontal lines

that are parallel. Alright, here's the next
step. We connect them, hence, the box so the

box gets made, right that you just call it
connect them. Alright, now I put a little

circle on the right side of the slide because
I wanted you to make sure you saw what's going

on there. Okay. That's when we put in q2 or
the median, right? So the median is 126. See

where 100 is. It's up a little bit, and we
make that parallel. But you see how I made

q one q three connected the box and then did
the median. I think this is the easiest order

to do it and when you're drawing it by hand
and you're not the statistical software Because

then that way, you know, this box is all nice.
And then your median fits and everything looks

nice, but we're not done yet. We got the whiskers.
So you're probably wondering this whole time,

what is this whisker thing? Well, you just
figured out what the boxes the whiskers are

the markers for the minimum and the maximum.
So you'll see the minimums at 41. And then

we have a whisker at 41. So why is it called
a whisker? Well, it's smaller. I don't know

why it's called the whisker, but it's different
from the other ones. Because it's smaller.

I guess that's a reason maybe. But notice
how it's like half the size, almost half the

size. Sometimes they're really, really small,
but it's tiny. And you want to position it,

like vertically in the middle, like you don't
want it off to the side or anything. But and

you also want these parallel. You'll notice
the maximums up there way high at 364. So

I just did both of these on the same slide.
So you draw on the whiskers. And then you

probably can guess the last step. Yeah, connect
the whiskers to the box. So good job. There

you went and did it You made a box plot. And
then now let's look at the inter quartile

range. Remember how you calculated this, you
took q three minus q one? Well, that means

this boxy thing is 164. Beds long, right?
So that's where your IQ are. This is a visual

pictorial of your IQ. So very good. We did
our boxplot, we did our inter quartile range.

And you're probably wondering, why don't we
just do this?

I'll explain.

So why do we do this? Well, one of the main
things that we do is we look at the distribution

in the data. I know, I know, you guys learn
how to do a histogram already, and you're

good at a stem and leaf. Those are other ways
of looking at the distribution. And if you

make a histogram of these data, you'll find
that Well, I mean, these are only 11. But

you know, if you get a pile of data, and you
make a histogram and the stem and leaf, you'll

find that those images agree with the boxplot.
And you're probably thinking, Well, how do

how do they agree? Well, if you look on the
right side of the slide, I'm just giving you

an example. So skewed, right? If you had skewed
right data, and you knew it, because you made

a histogram and you saw a skewed right distribution,
if you took the same data, and you made a

boxplot, it would be kind of like that skewed
right one that we just did, where the top,

whisker would be really high in that thing
connecting the whisker to the box. That would

be like really long, whereas the one on the
bottom is short. As you can see, the skewed

left is the opposite, right? The bottom one
is long, and the top one short. If you have

a normal distribution, remember that that's
symmetrical. That's that mound shaped distribution,

and you have a larger spread. In other words,
you have a bigger standard deviation, you

have a bigger variance, right? Then you're
going to see a box that's really big like

that. But if you have a smaller spread, and
it's a normal distribution, you're going to

see a box that looks like this. And you're
probably wondering, where are you getting

these shapes? Well, I'll show you a kind of
on the last slide here as we wrap up the conclusion.

It's because if you fly over a roller coaster,
like see this roller coaster, this roller

coaster is skewed right? That would make sense,
right? Because you want to go up steeply,

and then go down really fast. And see how
the boxplot for the roller coaster looks.

You've got sort of the part where you start
going up really fast. That's kind of near

the median and kind of near the the 25th percentile.
And the part where you start where you're

just getting on and it's slowly going there.
That's like the bottom whisker. And then you

go up and you come down. And it's a long tail,
which is good, I guess if you design roller

coasters, and then that long tail, then is
that right skew? So that's why I mean, if

in your mind, you're going how she getting
this this histogram in this box, but this

is kind of how I'm doing it, as I'm saying,
Well, if you flew over the histogram, or the

roller coaster, you might see like a shape
of a box plot. So in conclusion, we talked

about percentiles, in general, like the 77th
percentile, what that all means. And then

we focus in on quartiles, which are a specific
set of percentiles. And then we're going to

go or we already did calculate the quartiles.
And the reason why we did that is because

we first needed to do that in order to make
the interquartile range. And then finally,

we need those quartiles in order to make and
interpret a box and whisker plot. Okay, this

isn't the roller coaster I'm going to, but
I'm going to one and I guarantee you it is

skewed right.

Greetings and salutations. Hi, this is Monica
wahi, your library college lecturer bringing

to you chapter 4.1, scatter diagrams and linear
correlation. So here's what you're gonna learn

at the end of this lecture, you should be
able to explain what a scattergram is and

how to make one state what strength and direction
mean with respect to correlations and compute

correlation coefficient are using the computational
formula. And finally, you should be able to

describe why correlation is not necessarily
causation. So let's jump right into it. First,

we're going to talk about making a scatter
diagram. And the thing on the right side of

the screen is not a scatter diagram, but it's
kind of scattered. So I put it there, it's

kind of pretty. And then next, we're going
to talk about correlation coefficient, R,

and how to make it. And then finally, we're
gonna do a shout out to causation and lurking

variables, which remember we talked about
before, but we're going to talk about them

again, in relationship to our. So let's start
with the scattergram. And I also call it a

scatter plot, because it's like everything
in statistics, there's got to be about eight

names for everything. So scatter gram, and
scatterplot mean the same thing. So let's

just get with the setup here. So scatter grams,
or scatter plots are graphs of x, y pairs.

So what's an XY pair, xy pairs are measurements,
two measurements made of the same individual

or the same unit. So if you measure my height
and my weight, that's an XY pair, if you measure

my height in the my friend's weight, that's
not an XY pair, because that's two different

people, right? So these xy pairs, the x part
is called the explanatory or independent variable.

And it's always graphed on the x axis. So
remember, in algebra, you would do these graphs,

where you have this vertical line, and that
was the y axis, and you have this horizontal

line, which was the x axis. And I always had
trouble remembering, which is which, but that's

how it is. And so whichever x whichever of
the pairs is x, expect that to be graphed

along the x axis. And it's also called the
explanatory and or independent. Remember,

there's got to be a million names for everything
explanatory or independent variable. So if

I talk to you and said, here's an XY pair,
and this one is the independent variable,

or this one is the explanatory variable, you
need to like just secretly know I'm talking

about the X of the two. And then surprise,
here's the y of the two and the Y is also

called response variable. It's also called
the dependent variable. And that is graphed

on the y axis. So again, like I said, I used
to have trouble remembering is the vertical

one, the y axis or the horizontal one. But
what I did was I remembered, if you take a

capital Y, and you go grab onto its tail,
and you go pull it straight down, you'll see

that it's vertical. And that's how I remember
that's the y axis, it doesn't hurt the Y.

It's used to that. So if you can stretch the
y's tail down, and you get vertical, remember,

that's the y axis. And then the other one
is the x axis. Okay? And then also, you have

to find a way to remember which one means
what like, does x mean explanatory and independent?

Or what or does it mean response independent.
So how I do it is, you know how we sing the

ABCs abcdefg. Well, if you fast for the N
is w x, y, z, right, so the x comes before

the Y, you know, in the alphabet, so I do
x and then an arrow to y. And then I imagined

in my head that saying X causes Y, even though
it doesn't necessarily cause y's, you'll see

at the end of this lecture, but I think about
it that way. Because if that happens, then

y is dependent on x and x is independent,
it can do whatever it wants, but y is dependent.

So that's my way of remembering x is the independent
variable, and y is the dependent variable.

So anyway, that's a long way of saying the
scattergram is a graph of these xy pairs.

And that's what we're going to do is make
that graph. So we needed some xy pairs, right?

So I asked the question, do the number of
diagnoses a patient has, does that correlate

with the number of medications she or he takes?
So if you don't have that many diagnoses,

you probably aren't on that many meds, right.
But if you have a lot of diagnoses, you should

be on a lot of meds. But we all know

people in real life can sort of violate that
just depending, I mean, you could have one

really bad diagnosis with a lot of meds. Or
you can have a bunch of diagnoses that are

all taken care of with one mad so it's not
perfect, but this is kind of a reasonable

thing to think. So what I did was I put up
here just for x y, Paris, as you can see,

so I'm got four pretend patients. And you
can see here's the first patient, that person

has an x sub one because they only one diagnosis,
but like I was saying must be a bad diagnosis

because that person has a y of three or is
on three meds for it. Right? So that's how

you read this table. So let's start making
our scattergram out of these data. Okay, so

here we go. So I labeled the x axis number
of diagnoses, right just to keep things straight,

and the y axis number of medications, and
then you'll see where I put the dot, right?

because x is one, I went over to one number
of diagnosis, right? The one diagnosis, and

then, because why was three, I went up three
to this three, right, and there goes the dot,

that's where that first person gets a dot,
okay, you put it there. And that's what you're

going to do with these other ones, too, is
four dots. Okay, I just threw all the dots

down, so you can kind of see what was going
on. But here's the second person, right? So

that person had an X of three. So I went over
three. And I just put those green arrows in

just so you can see what was going on, they're
really not part of the scatterplot is just

more like, like cheating, you know, to show
you because we're just practicing right? And

then that person, so had an X of three and
then a y of five, and you see where the dot

goes right. And then here, you can see where
the fourth got.or I'm sorry, the third that

goes because there's a four and a four. And
then here we have the fourth that. So this

is the scattergram of these four patients.
Of course, a lot of times you have like hundreds

of patients in there. But I just showed you
the simple example. Okay, now, because we

did that, I can talk about linear correlation,
you'll kind of get it right. linear correlation,

that term means that when you make a scatterplot
of xy pairs, it kind of looks like a line.

Now over here on the right is not like biology.
That's not like statistics. That's like algebra,

right? Because back in algebra, you'd have
these perfect lines where the dot was right

on the line and see the x and y. Notice there's
no diagnosis, nothing. That's algebra, right.

So perfect linear correlation. Looks like
graphing points in algebra. And if you actually

make a scatterplot, of like people, xy pairs,
and you see that, you should suspect there's

something wrong, it actually happened to me
once, one of our statisticians came to me

and said, Monica,

look

at this, you won't believe this. And I said,
Well, I don't believe this. What are you graphing?

And he said, on the x axis, he had put the
weight of every of the person's liver. And

on the y axis, he put the weight of the whole
person. And I'm like, I, how do you weigh

people's livers? Like, that sounds painful.
And he goes, Oh, let me go see. And what he

learned was that you don't waste people's
livers, you use an equation to estimate the

weight of their liver and guess what's in
the equation is their actual weight. So I'm

like, that's why I came out, like on a line
is because you were using the Y to calculate

the x. And he was like, Oh, you're so smart
for a secretary. So then I became an epidemiologist.

But anyway, if you ever see this in biology,
just suspect Something's fishy, because really,

things just don't end up right on line. But
if they get really close, you can say it's

close to perfect linear correlation. I just
wanted to let you know, that's what we're

what's going on here with this linear correlation.
Okay, so let's talk about facts about linear

correlation. So things can be linearly correlated,
without being perfectly on the line, obviously,

our little thing was, so if, if when you make
those dots, your scattergram, if you imagine

a line going through it, if you imagine that
the line is going up, like it kind of looks

like it's going up, this is called a positive
correlation. But you don't always have a line

going up. So I want you to look at this. And
I made up these data too. But on the x axis

is the number of patient complaints. So as
we go on, the patients are madder and madder.

They're grouchy and gross, you're making more
complaints. on the y axis, we have number

of nurses staffed on the shift, right? And
so as you go up, there's more nurses. Well,

sure enough, when you got a lot of nurses,
you don't have as many patient complaints,

right? Because they're being attended to.
So this is what you would say is some people

say inverse correlation. But in this presentation,
I'm calling it a negative correlation. Because

as one goes up, the other goes down. And as
one goes down, the other goes up, because

and that's depicted visually with this line
going down so you see, you can imagine line

going down That's a negative correlation.
Neither is better, you know, positive versus

negative, it just explains how these things
are behaving together how X and Y behave together.

But then

you can have situations where there's really
no correlation, like x and y really don't

have anything to do with each other. So as
you've seen, you know, when you're, when you

have patients in the hospital, some of them
have really big families, and those families

come a lot. And some of them don't really
have that many loved ones. So as you can see

along x, here are totally unique visitors,
meaning you just count each person wants.

So you could have, there's a patient who only
has one Unique Visitor. But if you look at

why they spent in the hospital, that person
that's been there seven days, and that that

visitor keeps coming, right. And then you
have maybe a patient here, the second one

is to unique visitors. And that person's only
been in one day, but both those people have

been there, then you have people like a person
with three unique visitors. And they've been

in the hospital for days, right. And those
are probably the same three people coming

back. So it really doesn't matter how long
a person's in the hospital, if they've got

a lot of loved ones who keep coming, they'll
keep coming or not. Right? Right, according

to this correlation. So you end up imagining
a straight line. And that's no correlation,

that's fine, too. Nothing is better or worse,
it's just that you make the scattergram to

try and understand how x and y are related.
This is always fun. Like in books, they always

make some sort of goofy picture. I don't know
why they do this, I would never get a goofy

picture, like they show in books about, you
know, this, I made up the correlation. This

is in the lobby, the number of the games in
the lobby, and the number of the books in

the lobby, they should really have nothing
to do with each other. But if you see something

just way goofy like this, just say it's no
correlation. I don't even know how I get this.

Hi, there. Alright, so we've been talking
about correlation. And it actually has two

attributes. So far, we've only talked about
one and that is direction, we talked about

positive, negative and no correlation. So
whenever you're talking about a correlation,

you have to say what direction it is. But
you also have to say the other thing, which

is what strength it is. So now we're going
to talk about how you figure out what the

strength is. So strength refers to how close
to the line, all of the dots, they fall really

close to the line, it is considered strong.
If they fall kind of close to the line, it's

called moderate. And if they are very close
to the line is weak. Now remember, that's

totally different from what direction is it
could be positive, strong, or negative, strong,

right, could be positive, moderate, or negative,
moderate. So this is just a statement, the

strength is a statement of how close the dots
you make in your scattergram file close to

the line that you end up dropping. So I thought
I'd just give you a few examples. So look

at this, I just made this up. This is what
a strong negative one would look like. Notice

how those pink dots are almost on the line.
And this is a strong positive. Again, even

one of the dots is on all right, not all of
them, you know, or it'd be perfect, but it's

never perfect. So this is really close. But
it's strong, positive. So strong just refers

to the fact that the dots are almost on the
line. Now, this is almost the same correlation,

but the dots are not really almost on the
line has to be fair and kind of going between

them, but they're kind of far away. And so
just eyeballing it, you would say this is

moderate. And here, it gets weak. And mainly
it's because the dots are more all over the

place. But you'll notice there's one that's
like right on the x axis. And then hey, look

up there, like in the title, there's one up
there, like way up there. And that's like

an outlier. And sometimes, when you get outliers,
they can really whack things out. So even

though this is a weak correlation, that line
looks like so powerful, because it's almost

basically connecting these two outliers. So
you just got to be careful, and that's part

of why you make a scattergram first is out
large can have a really powerful effect on

the correlation.

Especially it's an any of the four corners
of the plot. Like if you get a weird outlier

kinda in the middle, it's not going to do
as much as if it's in the upper right, upper

left, lower right or lower left. It can really
affect the direction like like, you know,

it's Like a seesaw, or a teeter totter, you
know, an outlier can get on and really change

the direction of it. And it can also mess
with how strong or weak the correlation is.

So that's why you really want to start with
a scatterplot. And that's why the way this

chapter is organized starts with the scatterplot.
This, you just want to look for outliers.

And also just see how X and Y look when you
plot them. Now we're going to get on to correlation

coefficient, R, we're going to get on to computation
and actually making a number. So you can not

just use watery terms like direction, you
know, positive, negative, or moderate, strong

weak to explain it, but you can actually put
a number on how correlated x and y are. So

remember, the word coefficient, we did it
with coefficient of variation, which is different.

So the CV, you know, is one kind of coefficient.
But what we're going to talk about is a different

kind. This time, our coefficient, this time
is called R. And just coefficient means the

number we just like to use it in statistics.
Now, it seems kind of weird, because like,

I'm talking about correlation, and people
are like, Well, why is it our Why isn't it

like see for correlation, then like, I don't
know, I didn't invent it. But this is how

you can remember you can go correlation, correlation.
So correlation coefficient, R. So just remember,

r means correlation. And technically our mean
sample correlation, population correlation

coefficient, right? Like his, you know, imagine
you're correlating like height and weight

and the population like, oh, everybody in
particular state, you actually need a Greek

letter for that. And I showed it on the screen,
I don't know it's this fancy p, I don't know

the right name of it. But we don't actually
cover it in this class. So I just want to

just show it to you, we're only going to focus
on R, which is the sample correlation coefficient.

So what is r? Well, it's like I said, it's
the numerical quantification of how correlated

a set of x y pairs are. And it's actually
calculated by plugging all of the XY pairs

into the equation, I'll show you how to do
it. And you can see that if you do it by hand,

if you have a lot of xy pairs that will take
forever. So I tried to limit that. And like,

remember, standard deviation and variance,
there was like a defining formula and a computational

formula. This time, I'm only going to show
you the computational formula, it's, in my

opinion, ways your to do, but it gets you
the same number. Alright. So that's what we're

going to do is we're going to take a set of
xy pairs, and we're going to calculate

our

M. But then how do you interpret our Well,
let me just prepare you mentally for what

we're going to get out of this calculation.
The our calculation produces a number and

the lowest number possible is negative 1.0.
So that's perfect negative correlation. So

if we were like in algebra, and we had an
A line going down, and all the dots were on

it, then the R would be negative 1.0. But
that never happens. Right? So if you want

to think about it is like if you have a negative
correlation, and you get an R, that's like

negative point nine, five, or something really
close to negative 1.0, that it's close to

negative 1.0. So it's close to perfect negative
correlation. That's how you want to think

about it. And then the opposite is the highest
possible number you can get for our is 1.0.

But most people never do that. except for
that one mistake I was telling you about.

And that would be perfect positive correlation.
So if you see that you calculate an R, and

it gets really close, like point nine, five,
like I said, or nine, eight or whatever, then

you're thinking, whoa, this is really close
to perfect positive correlation, right? And

then everything else is in between. So like,
you know, point five or negative point three

or point 02, or negative point, oh nine, like
all of those are between negative 1.0 and

1.0. And that's where r should be. So let's
say you calculate R and you get eight. Okay,

you did it wrong, right? Or you calculate
R and you get negative 2.3. Like that's not

right, it's got to be between negative 1.0
and 1.0. And if you make a scattergram, you

should know whether it should be on the negative
side of the positive side or it should give

you a hint. So this is just more to calibrate
what to expect from our because it's kind

of a big calculation. So I'm just going to
give you some pictorial example. Because remember,

every single time we make our right, um, we
also have a scatterplot behind it. And I just

thought, you know, it would be helpful to
see some real life examples of our, these

are real life examples, okay, real life, you
don't get this from just anything, right?

I'm just teasing. But anyway, so I started
with some negative hours because I'm feeling

negative today. I went into the literature
and I found this article

about,

oh, it's not MIT and Harvard. It's about the
evolutionary principles of modular gene regulation,

a nice and all I know, it's, I'm supposed
to cut down on eating bread. So that's all

I know about this. But they had these really
nice scatter plots. So and they calculated

are for them, so and they had a little line
on them. So I thought I'd show them to you.

So if you look, the one that's labeled D,
see where the dots are, right, and see where

the line is. And this looks kind of like a
moderate to strong, negative correlation,

right? Because the dots are kind of close
to the line. And then when the group calculated

are they got negative point seven. And so
that kind of makes sense, because, and then

I put my opinion in the lower right, these
aren't official cut points or anything, but

I usually use these as a guide, see how I
said negative point four to negative point

seven is moderate. So I would call that the
one monitor. Now let's look at E. So see how

the dots don't cluster so close to the line,
as they do with the D one, that's going to

make it a weaker correlation, it's still it's
still negative, right? So it's negative point

four, four. And when you look at my little
opinion, I still call that moderate, but it's

on the low end, see that. And then if you
look at AF, see how many of them are like

way far away from that line, and they're dragging
it down. So now it's in the even weaker correlation,

negative point two, five, right. And so then
that's weak. And so this is just some examples

to give you a pictorial. And now I'll be I
promise to be more positive, here's some positive

Rs, they didn't draw a line on this one, this
is a different article, right? Says obesity

is associated with macrophage accumulation,
and adipose tissue. So again, try to cut down

on bread. But anyway, um, if you look on the
left side, you'll see all of these x y pairs

plotted on the scattergram. And even though
we don't have a line there, we can imagine

it's going up. So we would expect this to
be positive. But we also would imagine they're

not really clustering around the line very
tightly. So when we see that the R is point

six, we're not surprised. I mean, it's on
the high side, a moderate in my world, which

makes sense. But go look on the right one,
you know, under the B one, look at how those,

you could almost connect the dots and get
a line out of that. So that's really tightly

hugging the line. And then we're not surprised
to see that the R is point nine, two. So that's

pretty strong. So I just wanted to give you
these tutorials before we actually went forth,

and calculated r because that's one thing
you can do is do the scatterplot have an expectation,

what r should look like. And then if you calculate
R and it's totally wacky, you know that you

did something wrong. Okay, let's calculate
our and let's use the computational formula.

Okay, I threw the formula up in the upper
left, and don't feel overwhelmed by it, we're

going to take that apart very carefully, right.
But before we even do that, I just want you

to have a flashback to chapter 3.2. c, all
those sums of are those capital sigma was

in the equation. So we're going to handle
calculating are a lot like we handled calculation,

calculating variance and standard deviation.
We're going to make like a table with columns.

And then we're going to fill in those columns
with calculations. And then we're going to

add up the columns to get all those numbers.
So already you were good at that, and 3.2,

you'll be good at this too. And then I made
up a story because it's a lot easier to check

your work if there's some story behind that
and statistics. So pretend we have seven patients

that have been going to your clinic for a
year. They're good patients, they keep coming.

So they came to the clinic over the year.
And at the last visit of the year. You measured

the diastolic blood

pressure, and what you predicted was or what
you thought would make sense as those with

a higher diastolic blood pressure would have
had more appointments over the year because

probably they're trying to stabilize and run
power. Sure, maybe they have other problems

that are driving it up. This makes perfect
sense, right? So what you wanted to do is

see if you are right, so you're going to take
the diastolic blood pressure at the last appointment

as your x, you know, because you think that
that's maybe the explanatory variable, or,

you know, that would be the independent variable
that would make it so have something to do

with whether or not they had a lot of appointments.
And then you take why as the number of appointments

over the last year, because you'd say, Okay,
hi, DBP probably means they have more appointments.

That's just your idea, maybe you're wrong,
but we're gonna do that. Okay. So, um, I put

in the title, just a reminder, access DVP.
And why is number of appointments so you don't

forget. And then we made up this tape. So
look at the first column, it's just the patient

number, it's nothing, you know, exciting,
we just want to keep track of which patient

is one, right. And then notice under x, we
just have all of their dbps. So this patient,

one at the last appointment had a 70 mmHg,
and patient two at 115, mmHg. That's

kind of alarming.

But these are fake data. So don't get worried
about these patients. But anyway, we just

fill in x. And then also, when you have their
chart out, you can look up how many appointments

they had over the last year and patient went
only at three, whereas patient two had like

45, which you can believe because sometimes
they're coming in all the time to get stuff,

adjusted. It but then you know, patient three,
only a 21 and patient four at seven. So you

can see these are the XY pairs for each of
these patients, right. And it's pretty simple

to go to the bottom and sum up each of the
columns, we have some of xs 678 and some of

y's 166. And also, I'm reminding you of the
our calculation, I put that in the upper right,

just so we can see what we're doing. I just
want to call your attention to one of the

terms in there, which is sum of X, which I
put in the parentheses here. And that we already

know, just from making the first part of this
table and adding it up. So we already have

that thing. And now I just wanted to point
out, if you saw the sum of x over here, it's

not exactly the sum, it's a sum of x y. So
the Y is mushed. Right next to it, that's

not some of x, that's some of x y. And that's
later in the game, we're gonna put the sum

of x y at the bottom of the last column. So
So that first term there, that's not some

of x, that's some of x y. Okay, now downstairs,
we see the sum of x to the second, right?

And that looks an awful lot like the one next
to it on the left, which says sum of x to

the second, right? And so how do you tell
the difference between the kind without the

parentheses and the kind with the parentheses.
So this is how I do. The rule is always regardless

what's going on, do what's in the parentheses
first. So that's easy to do. If you have parentheses,

if you got the parentheses version, you know
that the sum of x to the second with the parentheses

in it, is you just do the sum of X, and you
do the sum of X and E times by each other.

Right? But what if you don't have any? Well,
what I do is I say, Well, if I did have some,

I do it this way. But if I don't have any,
then I know I have to do the sum of the x

squared calm, right. So that's where you take
x times x x times x, x times x on each line,

put it there and sum that. So that's how I
go through it no matter where I am in statistics

or algebra. If I see that some symbol and
then the x squared, I first look for the parentheses.

If they're there, I know what to do. If they're
not there, then I know you don't do the thing

where you just take the sum of x squared,
you have to go and look at the bottom of the

column of the x to the second column and take
the sum of that. I hope this is helpful. All

right, so as you can see, there's, I've shown
you on the top of the equation is where you

just take the sum of X and the sum of Y. And
on the bottom, I'm showing you where you take

those and you take the square of them. And
then in the other term is the one where you

just take the sum of the call. All right.
And so there you go. So what happened here?

Well, we filled an x to the second so if you
go to a patient, 170 times 70 is 4900. That's

where we're getting that number. So you go
through and then patient to 115 times 115

is 13,225. So you go from All those and then
you sum those up. And that's what goes in

that first term. And then I'll bet you can
guess what the next

slide is.

Surprise. Now we do the y one, so don't get
confused because you kinda have to skip a

column there. So three times three is nine.
And so that's why in the Y squared, I'm 45

times 45 is 2025. That's how we're doing those.
You sum all that up, and then go look up at

the equation, that's where you put that sum
of Y squared. Now we have x, y. And this reminds

me of a student I had before. She was really
confused. She's like, Monica, I don't know

what to do with x, y, the x, y quantity. And
I go, What do you mean? I mean, it's pretty

obvious. You just take x times y, like here,
70 times three is 210. She goes, x times y,

where's the times? Like, how do you know it's
supposed to be times like, I don't see any

times. Right? I don't see any dimes either.
Like there's no like, like, how do you know

to do that? Well, anyway, I'll just tell you,
I guess, imagine, like a little multiplication

symbol between x and y. That's what's supposed
to be there. That's what you're supposed to

imagine, I guess I was so used to looking
at it was like, you're right, I guess you're

just supposed to assume that. So take x times
y. So for patient two, we just took 115 times

45. And that's how we got 5175. So you go
through each of those, it's a lot of processing.

And then you sum it up at the bottom, whoo,
that's a big number. And then you see, I circled

it in the our equation. So I think we figured
out where to put everything, obviously, n

is seven, right, because we have seven patients,
you see a bunch of ends in there. So I think

we have all our ingredients. So let's move
forward. So all I did here was rewrite the

exact same equation with all the ingredients
in it, right. So like I said, the N is seven.

And so wherever you see n, you'll see a seven.
See that sum of X, Y on the top, you see where

that goes, see some of x and some of y and
then downstairs, you'll see I filled in all

those numbers too. Now, let me just talk to
you a little bit about both levels, the numerator

and the denominator in the numerator, because
we have order of operation, you need to do

out the end times the sum of x y, that's seven
times 18,458, you need to do that out first.

And then you need to do the other one, you
know the 678 times 166 first, and then after

you're done with those two things, you have
to subtract the second one from the first

one, that's the order you have to do that
in to get the numerator right. Now for the

denominator, it's a little bit the same, but
a little more complicated. You see on the

left side, you have that seven times 67,892,
you have to do that out. And then you have

678 squared, you have to do that out, then
you have to take that, subtract it from the

first one. And after that, after you have
that, you take a square root of all of that,

and that's your first term. And then you still
have to go over to the other one, you have

to take seven times 6768. Keep that then take
166 times 166. Keep that, that that term,

you subtract from the first one. And after
you're done with all that, you take the square

root of that, and then those two things, you
have to multiply together. So that's a lot

of work, and you have to do it in the right
order. So here, I just wanted you to see how

you, you probably want to just work out this
term separately first, and then work out this

terms separately. And just like that thing
I was telling you about x y, those two terms,

once you work them out, you take the square
root of the left one in the square root of

the right one, you have to multiply them together
to get the denominator. So this slide is to

help you see I threw the numerator on that
was relatively easy. But these are the two

different numbers you should get from the
left side of the denominator and the right

side of the denominator just to check your
work. And then of course, once you multiply

them by each other, you get this number 17,561.3.
So ultimately, what the calculation for our

comes down to is you're trying to calculate
the numerator and you're trying to calculate

the denominator. And at the end, you divide
the numerator by the denominator and you get

the answer which is R. So we're going to do
that now.

And here's what we got is we got this 0.949.
And because we see that it's positive, then

we know it's a positive correlation. And then
remember my opinion. And also probably everyone's

opinion, because if you run that up, you go
point nine, five, well, that's getting really

close to 1.0. So most people would agree that
that's pretty strong. So how you would diagnose

this correlation is you would say it's positive,
and it's strong. Okay, I just want to wrap

this up by giving you a few facts about our
that I may not have covered yet. First, r

requires data with a bi variate normal distribution,
which is something we didn't check before

doing our r in this class, because I just
don't cover that. But please know, if you

take another statistics class, and they bring
up our, they might talk about checking for

the by various normal distributions. So just
know about. Next, please know that our also

does not have any units. So other things that
don't have units, remember, the coefficient

of variation didn't have any units, some things
just don't have units, and r is one of them.

Also, we did talk about how perfect linear
correlation is where r equals negative 1.0.

That's if it's a negative correlation, or
r equals 1.0, which is a positive correlation.

But I might not have mentioned that no linear
correlation is r equals zero. Now, you probably

won't see that in real life. But sometimes
I'll make an R, and the R is either positive

or negative. But it's 0.0000000. Something
right? Regardless of whether it's positive

or negative, if it's 0.00000, something, it's
really close to zero. So that means there's

probably like, no linear correlation. And
then we learned about positive or negative

art, but I just wanted to remind you of the
behavior of X and Y when you get those circumstances,

okay. So if you have a positive R, it means
as x goes up, y goes up. But it also means

as x goes down, y goes down. So they travel
together. When you get a negative r, it means

as x goes up, y goes down. But also it means
opposite, as x goes down, y goes up, so they

travel in the opposite directions. Now, here's
another fact about our little factoid, if

you choose to switch the axes, like let's
say I designate, you give me xy pairs, and

I designate a certain variable as x and the
certainly one is y, and you actually designate

them the opposite, it really doesn't matter
even in the equation, because you'll end up

with the same R value. So it doesn't matter
if you call the x my X, Y, and I call your,

you know, y x, like we can switch them, but
you'll still end up with the same are with

the calculation. Then finally, even if you
converted x&y to different units, you get

the same error. So let's say that you were
in England, and you were doing the correlation

between height and weight. And you were using
the metric system on the same patients that

I was using the US system, even though we'd
have different numbers, cuz obviously you

have to convert them, we'd still get the same
are when we're done. So finally, we get to

the last subject of this lecture, which is
lurking variables, which you've heard about

before. But the main point I want to make
is correlation is not causation. So you don't

want to be misled

by correlations.

So beware of lurking variable. So remember,
lurking variables are things lurking behind

the scenes, I caused things, right. And so
you may have realized that selecting x and

y, like if you have xy pairs, designating
which one is x and which one is y is kind

of political, because you're implying that
x could cause y. So let's say that you're

correlating height and weight, taller, people
are heavier. So you would cause x to be height

and y to be weight. You know, people don't
go, Oh, I'm too short, I should gain weight

so I can grow taller. You know, that's just
not the way things work. So you have to put

x as the height, and y is the weight. But
there are Riya. In reality, other causes of

weight besides height. In fact, there are
things that cause both height and weight,

like genetics, right? So a genetic profile
that leads to Thomas and also obesity could

be a lurking variable in the relationship
between height and weight. So there could

be some tall people that are always obese,
and it's not really just because they're tall.

It could be because they have the genetics
that programmed them to be tall and also obese,

right? And so here's an example where you
got to be real careful. Um, with correlation.

So there's been this claim that eating ice
cream causes murders, because they noticed

when in areas where ice cream sales go up,
murder rates rise. And I don't know about

you, but when I have some really good ice
cream, it just makes me so mad. I'm just kidding.

I mean, why would this happened? Right? Well,
the reality is summer and warm weather are

lurking variables, because we sell more ice
cream in the summer. You know, the ice cream

consumption goes up. But also people are outside
more and more murders occur. And you know,

I from Minnesota, where it gets really cold
for periods of the winter, and oh my gosh,

there are totally no murders, then, like people
just don't commit murders, when it's really

frigid out, it's just really inconvenient.
So that's a situation where there's a lurking

variable. And so you don't want to start,
you know, screwing up our ice cream laws and

making it so we can have ice cream, just because
you misappropriate that ice cream causes murders,

right? There's a lurking variable behind it,
that's having something to do with both. Here's

another one. And this was my professor in
my biostatistics class, they use the C put

up a really like a time series chart over
a long time, like since the 1900s. And they

pointed out as people purchase more onions,
the overtime is onion consumption goes up

and down. The stock market rises, right? So
when the stock market slow, people aren't

eating as many onions. And this is just true
over generations in the US. So um, yeah, we've

had some problems with our economy in the
US, do you think we should all start eating

a bunch of onions, right? So the healthy economy
is a lurking variable. And a healthy economy,

people buy more food, they including onions,
and also a healthy economy boost the stock

market. So you got to be careful about this
correlation is not causation. You know, and

so if you want to make the stock market go
up, don't make everybody onions. And definitely

don't make a stop eating ice cream, that would
make me very upset. So at the end of the day,

you're not going to be able to affect the
murder rate by bringing down the ice cream

consumption rate. And you're not going to
be able to fix the stock market by making

people eat onions. And so that's the whole
concept behind lurking variables. And correlation

is not necessarily causation. So in conclusion,
when you're doing your correlations, First,

make a scattergram because you want to get
an idea visual idea of the strength in their

direction. And you also want to look for outliers,
then go on and calculate are by hand, but

be really careful because it's a big hairy
calculation. And you don't want to make any

mistakes. And then finally, when you go to
interpret are Be careful of lurking variables.

And remember that correlation is not necessarily
causation. And now, time for some ice cream.

Hello, it's Monica wahi, your library college
lecturer here to ruin your day with chapter

4.2 linear regression and the coefficient
of determination. So at the end of this probably

painstaking lecture, the student should be
able to at least explain what the least squares

line is. Identify and describe the components
of the least squares line equation, explain

how to calculate the residuals, and calculate
and interpret the coefficient of determination,

or CD for short. Alright, so it's really cool
if you have a crystal ball, because then you

can make predictions, right, you just look
into the crystal ball. It's some nice equipment,

I've had friends who have them, they're very
nice to put out on your dining room table

as the centerpiece. Unfortunately, though,
they don't really play much into statistical

prediction. So what I'm going to show you
in this lecture is how we use statistics for

prediction instead of this beautiful crystal
ball. So we're going to start by talking about

what the least squares line is. And then we're
going to talk about the least squares line

equation, which is the crystal ball thing
we use only in statistics, okay. And then

we're going to talk about dealing with prediction
using the least squares line. And finally,

we're going to talk about the coefficient
of determination. So let's get started. And

let's get started with the term least squares.
criterion, right? So remember, criteria is

plural and criterion is singular. And it means
well criteria as stuff you need to meet right

to be eligible like you have to meet the criteria
for registration for college right? Well,

least squares Cartier tyrian is just one,
which is awesome, because then you only have

to meet one thing. So one of the things you
probably wondered when you were watching last

lecture is how do you know exactly where to
draw this line when you have a scatterplot.

Like, how do you know where to make the line
the most fair. So in the last chapter, when

we plotted the scatter grams, I just drew
a line there for demonstration. But there

actually is an official rule as to where the
line goes. Okay. And basically, the rule is

as has to meet the least squares criteria.
Okay? if it meets that criteria, there's only

one line that does, then that is where the
line goes. So how do we

get to that?

Well, this is roughly what it looks like.
When you draw the line, there is a vertical

distance from each of the dots to the line.
Now, as you can see, by the slide, sometimes

the dots are below the line. And sometimes
they're above the line. And so the word square

is indicates that whether it's up or down,
you're going to square it. So it's not going

to be negative anymore. Because whenever you
square a negative, it becomes positive. So

first, you're going to have to square all
of these things. Okay? So imagine you were

just going to try it out, like, maybe draw
this line, and then you calculate the squares,

and you'd be like, okay, that's how many and
then maybe you tilt the line a little. and

calculate the scores again. And your goal
would be to add when you added up all the

squares, to have the least ones. So the line
belongs where what causes smallest sum of

squares for the whole data set. So if your
software, which you're not you're a person,

right, but if you were software, you'd be
figuring that out using your software brain

as well, how exactly to tilt this line, and
where exactly to put it to minimize these

squares, but we're people. So I'm going to
go on and explain how people do this. So the

trick is, if you can figure out with the line
close, you can draw it on the scatterplot

and be right. But there is a challenge of
knowing exactly where it belongs on the graph.

And then also, you're probably realizing you
don't always have a graph to draw it on. Like

maybe you need to talk to somebody about where
the line goes, and you can't draw a picture.

So how you explain where the line goes as
you use an equation. And some of you may remember

this, and some of you may not, so I thought
I'd do a little quick review of how lines

and equations relate. Okay, so we're going
to get into the least squares line equation.

But first, I'm going to give you a little
flashback about algebra, and I'm sorry, if

this is painful, um, this is hard for me,
because I wasn't really that good at algebra.

But um, I and this isn't statistics, this
is algebra, but I just wanted you to remember

this part. Okay. So back in algebra, there
was a chapter, where you were given these

xy pairs, and then was different from statistics,
because they all lined up on a line, see,

these pink dots are just perfectly out of
line, okay, and these are the XY pairs. And

remember, you had to graph this kind of like
we had to do scatter plots. And then you were

given this equation, y equals b x plus a,
right? And that was the linear equation to

describe this line. And you were like, okay,
I don't get how to put this equation together

with this line. And so first, the teacher
would say, well, B stands for the slope of

the line, right? Because you have to know
the slope, I mean, the line can be tilted,

any which way. And so if you know the slope,
you already know something about the line.

And in algebra, how you would make the slope
as you calculate the rise over the run, right.

And so there, you know, be in algebra was
rise over run, and you'd get the slope. And

then you'd be like, great. But you'll always
needed another thing in order to define the

line. Because if you imagine this line is
in an elevator, it could still have the same

slope, but go up or down, right, so we need
to anchor it on the y axis somewhere. So h

stands for the Y interceptor where it's Spears
through the y axis. And, as you can see, by

the drawing, it looks like a is zero comma,
zero, right? But you don't have to look at

it, what you can do in algebra, is you to
get a is what you would do is go since you'd

filled in B, you just go grab an XY pair,
and plug the X and and plug the y and then

plug the B, you just got in and back. Calculate
the y intercept, right. And that's how you

would get the whole linear equation. And so
that's how you would do it in algebra. And

I just wanted to remind you that because we
do some similar things in statistics, it's

a little different. But I wanted to remind
you how to connect what a line looks like

with how this equation works. All right. Well,
welcome to statistics looks, those pink things

are not on a line. So we want to make a line
but now you know about the least squares criterion.

What you're trying to do is make a line that
minimizes the least squares, right? So here

we go. Um, remember Hello, I was just talking
about this linear equation back in algebra.

Well notice the difference. The main difference
here is the hat, right? The y is wearing a

hat. And that's universally in statistics,
whenever you see a letter or a number wearing

a hat, it means it's an estimate. Okay? So
of course, we're estimating why because if

you look on that line, none of these dots
actually falls

on that line.

And we don't really expect even an estimate
to fall on that line just close, right? You

know, because of the least squares, okay.
And so we almost have, in a way, the same

goal we did back in algebra, we have to get
that be that slope. And then we have to use

that to back calculate our a. Okay, so let's
go on with that. Um, so like I said, in the

software approach, you just feed all the XY
pairs in, and then the software just actually

prints out the B in the A, it just prints
out the slope and the y intercept, which is

why I love the software. But we don't get
to use that in our class. In our class, we

have to do the manual approach just because
it's painful. And I had to do too. So now

I'm making you do it right, me. Okay, what,
what we'll do is plug all the XY pairs into

an equation to get the slope, the speed. And
I promise you, I won't give you a ton of xy

pairs, you know, or you'll be there forever.
But this next step, we have to do, we didn't

have to do an algebra. And that is we're going
to have to go back to all of our x's, calculate

x bar, and go back to all of our y's and calculate
y bar. Remember, that's the mean of the x's

in the mean of the y's. And you're probably
wondering, Well, why do we have to do that?

I'll show you again. But in case you didn't
notice, though, those dots really didn't fall

on least squares line, they fell around, and
you need a.at least on that line to help back

calculate that wider set. And the rule of
the least squares line, one of the rules of

it is that x bar comma y bar is on that least
squares line. So you can know if you calculate

that out that that's actually on the least
squares line. Okay. And so finally, after

you do x bar and y bar, you plug in B, and
you plug in x bar for the x, and you plug

in y bar for the Y hat to back calculate the
A. So it's a similar, but different process

as algebra. So the moral of the story is you
need to recycle, right, we got to be good

to the environment. So what has happened?
Well, you wouldn't be at this point in your

life of making a least squares line, if you
hadn't already started out by making a scatterplot.

And then deciding you wanted to do R, and
then making are. And when you make Are you

end up with that big table, remember, and
you end up with all these calculations, like

some of x, some of y, some of x squared and
some of x y. Now you want to recycle those,

you want to save those calculations from our
because they fit also into the equation for

b. So you want to recycle that. Also, you
want to save the are you made, because you're

going to recycle that into the coefficient
of determination, which I'll explain later.

And then this is not about recycling, you'll
actually have to make this a new, but you

need to calculate x bar and y bar. Now you
never needed to do that before now, but now

you need this. And so yeah, so get together
your old r calculations, and then put your

x bar and y bar together and you'll be ready
to do the least squares line equation. Alright,

so here's a flashback. Remember this big table?
Remember our story, we had seven patients,

right? And x was their diastolic blood pressure
at

the last

visit they had of the year. And then why was
the number of appointments they had over the

year. And we thought, Well, if your diastolic
blood pressure, you know goes up, then maybe

you need more appointments because it's marker
of being sick. I don't know. That was my little

story. Okay, so over on the right now we'll
see that the formula, we have the formula

we're using for B, the tax gives you two formulas,
again, I've always got my favorite, it's the

one with the table, right? So here's the formula
for B. And then after you calculate B, you'll

notice in the formula for a, b is in the formula
for a so you got to do B first, right. So

a lot of times students are a little confused
and what the goal is here, the goal is to

if you look at the bottom of the slide, the
goal is to come up with what B is and what

A is, and then fill it in. And that's your
least squares line equation. So your least

squares line equation is always going to have
an A y hat in it. That's that's a variable

that just gets to stay there. It's always
going to have that equals and then after that,

whatever your B is going to be mushed up next
to that x so it's always gonna have that x

there. And then plus and then whatever you
get for a and just as a trick, if a Turns

out to be negative, then it ends up being
minus a, right. But that's the generic equation.

And our goal is to calculate B and A and fill
them in. And then we will say this is our

least squares line equation. Oh, remember
how I was saying, you actually need to make

some new calculations, right. So you need
to make y bar and you need to make x bar.

And it's a little easier to show when I've
got this column, the columns up. If you look

at the bottom of the slide, remember how some
of X was six, some D eight and remember how

our n is seven. And remember how a sum of
x divided by n is your x bar. And the same

goes for y, right, we have the sum of Y divided
by seven, I just wanted to quickly remind

you of this, that you need to generate these
things before, you can actually completely

finish the least squares line equation. I
just summarized like that I cut to the chase,

basically, I just summarize the the actual
numbers you're going to need and put them

over here. So we don't have to look at that
whole big table anymore. Alright, and you'll

notice that I grayed out the sum of Y squared
because I realized later we don't really use

that. Okay, so let's look under on the left
side under the big list of numbers we have.

And you'll see the B equation that I filled
in, right, and if you compare that to the

formula on the right side, you'll see what's
going on, you know that n is seven, right?

So wherever you see that seven, that's where
n is okay, then the top of equation, remember

some of x, y, let's just look that up. Yeah,
that's that big number 18,458,

I wanted to just be clear, you have to do
out that left side, the seven times the 18,458,

you have to do that one out, and then do out
the right side, which is that sum of x times

sum of Y which is 678 times 166, you have
to do that one out. And then after that, you

have to subtract the right one from the left
one, because of order of operation. Okay,

so that's how you make the numerator. Now
let's just look downstairs, again, we have

an n, so we know that's seven, and then that
sum of x squared. And remember, it doesn't

have the parentheses around the sum of X square,
if it had the parentheses around it, you'd

be taking like 678 and squaring that, but
it doesn't have the parentheses. So you have

to use that big numbers 67,892. Okay. And
again, like with the upstairs, you got to

do out that side of the equation, right, that
term, you've got to multiply that out before

even looking at the rest of the equation,
right. And then Oh, here we go. On the right

side of the denominator, we have some of x
squared, that's exactly the example I was

giving earlier. So you say 678 times 678.
And you have to do that one out, right. And

then after you do that one out, and you do
the first one out, then you subtract the second

one from the first one, remember order of
operation. And if you do it right, you should

get C below the on the left side of the slide,
you should get that for the numerator in that

for the denominator, and then you divide them
out and you get 1.1. And that's your B, right.

So there you go. That's how you do it. And
so now we got to worry about AES. So what

I did was I just wrote B at the top there,
so B is 1.1. And so now we can use B to try

and figure out a, so remember how I look at
my list. Remember, I did x bar and y bar for

you just so we had that ready. So now we're
going to calculate a by putting in Y bar minus

and remember order of operation again, we
got to do the B which is 1.1 times x bar.

So we do that one out first, and then subtract
it from 23.7. And remember, remember, I was

saying sometimes you get a negative a, well,
we got negative ad for a. Alright, so we got

our B, we got our a, and let's go. Now, oh,
if you want to check your work, this should

work out right. Like you should be able to
take the B times the x bar, right, which is

1.1 times 96.9 minus 80. You know the a and
you should get 23.7. So if that works out,

then you know you did everything right. But
remember what the goal was, the goal was to

actually fill in that least squares line equation.
So if you look over on the right, that's what

we did. So we still have our Y hat, we still
have our equals, now we have a 1.1 where the

B belongs. We still have that x because those
are variables that we had in the x, and then

we do minus 80. Because we came out with a
negative one. If it had been just plain 80

would say plus ad, okay. All right at the
beginning of this presentation, I teased you

that we were going to do prediction with the
least squares line equation. We weren't going

to use a crystal ball. We were going to Use
this equation. Well, I finally get to that

exciting part of this presentation. But, and
there's always a big, but I first have to

warm you up with some rules, right? First
of all, I just want you to reflect on what

we just did. And realize that we can draw
the least squares line. But unlike algebra,

our xy pairs probably aren't on it, right?
Like in this example, none of the XY pairs

are on it. So you need to be sure about at
least one xy pair that's actually going to

land on the least squares line. And the only
one that you can be sure of is going to land

on least squares line is x bar, comma y bar.
And if you reflect on it, that's why we had

to calculate that right, because we had to
use x bar and y bar in the calculation to

back calculate a the y intercept. Now, you
may be lucky and get a data set that there

is an x y pair that just happens to fall on
the least squares line, or maybe even a couple

or maybe more. But you can't trust that. So
if you need to trust that there's a point

on the least squares line, you know, it's
always going to be x bar comma y bar. All

right.

And now I want to focus more succinctly, on
to the slope or B, right. So remember, we

just in our example, calculated B and we got
1.1. For me, and that's a slope. So I want

to point it out that the slope B of the least
squares lines tells us how many units the

response variable or Y is expected to change
for each one unit of change and the explanatory

variable or x. So that's a little kind of
a tongue twister. But if you think of our

example, it's a little easier to understand.
So the fact that that slope was 1.1, in our

example, and that we were having XP DBP. And
why be number of appointments over the last

year, what we're essentially saying by that
is, for each increase in one mmHg of DBP,

or the X for each increasing one of those,
there is a 1.1 increase in the number of appointments

the patient had over the past year. So as
DBP goes up by one, then the appointments

goes up by 1.1. Well, I don't know what 1/10
of an appointment is, but you get what I'm

saying because it's just a Y, okay. And so
the number of units change in the Y for each

unit change in X is called the marginal change
in the Y. So which if you sort of think about

it, that's 1.1. So 1.1 is the slope. But 1.1
is also the marginal change in the Y for each

unit change in the x. Now, I also want to
just recall for you this concept of influential

points, right, so like with our if a point
is an outlier, and remember, we should have

done a scatterplot. And everything before
we got to this point, because we need our

we need all those sums of x's and sums of
y's and sums of sums and whatever, right.

And so like with AR, if a point is an outlier,
and you can see it on the scatterplot, it

can really drastically influenced the least
squares line equation, just like it's can

screw up our right. And so an extremely high
x or an extremely low X can do this. And I

was just, you know, pointing out a culprit
we have here on the scatterplot. So always

check your scattergram first for outliers,
because you could end up in a situation where

you're making a least squares line and there's
a bunch of outliers, you know, whacking it

out. Okay, now I'm gonna also bring up, you're
probably like, when do we get to the prediction

part? I'm like, you just have to relax, I
have to get through a few of these issues,

right? So one of them is the residual. And
you know, the word residual, like it kind

of sounds like residue, right? Like you said,
you know, somebody comes over and sits there

their cup on your coffee table without using
a coaster that leaves some residue and you

get all mad, okay, well, that's kind of what
a residual is. It's like kind of like residue,

it's like something left over, right. So once
the equation is there, once you make the least

squares line equation, there's something I
just want you to notice. And that is you can

take each x, remember how we had seven patients,
they each had an X, you can theoretically

take each x, plug it into the equation and
get the Y hat out, right? So I want to just

demonstrate doing that. So we have our equation
upper right here. So a patient one, I took

patient ones x which was 70. And I plugged
it in 70 times 1.1 minus 80. You know, I put

in the equation and I got negative three.
Now that's why had the real why I put it on

the screen here is actually three. So as you
can see, you know it's not the same answer,

right? And then patient two I did it with
patient two also I did 1.1 times 115 because

that's the x and then minus 80. You know,
because that's the rest of the equation. And

I got 46.5 Now that was a little closer, because
look at patient twos wise. That was 45 If

it's really close to this 46.5, that's a little
bit better. But the reason I was doing all

that is I just wanted to tell you the residual
is y minus y hat. So in the first case, we

have y hat was negative three and y was three.
So patient when we did three minus negative

three, and we got sick, so that's the residual,
it's kind of like residue, right? It's like

the residue leftover between Y hat and y,
right. And then patient who we did it again,

we took y which is 45 minus y hat, which was
bigger, it was 46.5. So we got negative 1.5.

So that's the residual. So So this is how
you calculate the residual. And this is what

it is, this is how you get it. But the bottom
line is, you don't want big residuals, right?

Because that would mean the line didn't fit
very well. So you'll find that if you have

a really good fitting line, you have very
small residuals. And so you're probably like,

well, what's a good fitting line? Well, we'll
get to the coefficient of determination, and

that'll help you see what constitutes a good
fitting line.

But first, I will get to the prediction part,
okay. So you're done with your least squares

line equation, and you want to use it for
prediction. So let's say you knew someone's

DVP, and you wanted to predict how many appointments
she or he would have in the next year. Now,

what you're not doing is you're not using,
you're not reusing your X's from your data,

we just did that to make the residuals, what
you're doing is actually imagining a new thing

out there. And you're gonna use this equation
for prediction. So you could plug in the DVP

as an X, and get the Y hat out, and say that's
your prediction, right? But you gotta use

some caution. If you use an X within the range
of the original equation, as you can see,

I put the x's up here, the range of the original
equation was like 70 to 125. Right, those

were, you know, the areas covered by x, right?
If you do that, if you pick an X, somewhere

in there, this type of prediction is called
interpolation. And people feel pretty good

about it. But if you use an x from outside
the range, like one that's really smaller,

like 65, or one that's bigger, like 130, then
it's called extrapolation. And then it's not

such a good idea, because you don't know if
it's really going to work, right. So here,

I'm going to give you an example of interpolation.
The patient in your study as a DBP of 80.

Okay, so 80s, right in there, it's in that
range. So let's use it right. So we do it.

Now, this looks familiar to you, because we
just did this when we did residuals, but we're

using a new person now. So 1.1, times 80,
minus 80, equals eight. So this is how we,

what we would do is predict that this patient
would come to eight appointments next year.

So there, that's how we use our least squares
line equation, like a crystal ball where we

can predict right? So is it really this easy,
right? Is this all you have to do to predict

the future? Well, it's not really that easy.
You can't make a linear equation out of any

old xy pair. So remember this from our last
lecture, see, the scatterplot. It looks like

what a cloud in That's right. It doesn't have
a linear equation, you know, it doesn't look

like it should make a line. But you know what,
you feed that stuff into the software, or

you feed that stuff into your B formula, and
you're a formula, you'll get, you'll get a

line out of it, even if there's no linear
correlation. And so if you get that line out

of some scatterplot, that looks like this,
then it's not a very good line, right? And

it wouldn't work very well for prediction,
right? Because this looks pretty unpredictable.

So for that reason, we can't just accept any
line that is handed to us. To evaluate if

our least squares line equation should be
used for interpretation, we need the coefficient

of determination. So here we are at the coefficient
of determination. And so remember how I said

you have to recycle, recycle recycle in this,
well get out your our time to recycle. So

the coefficient of determination is also called
r squared. And it literally means r times

r. And I just have to add this on. Just like
remember the coefficient of variation. Remember

that one, we always turn r squared into a
percent, right? And so you times it by 101%.

So in this example that we did remember, early
on in the last lecture, we did the R for this,

that not the scatterplot I just showed you,
but the for the one of DBP, and the appointments,

right? And we got an R that was really, really
strong positive correlation, right, we got

point nine, five. Well, if we want to calculate
r squared, which is the coefficient of determination,

we take point nine five times point nine,
five If and we get point nine oh, but we got

to do that percent thing. So we end up with
90%. So this is how you say it, you say that

90% is the variation that's explained? And
why, by the linear equation, right? So that's,

you know, y varies, right? Like how many appointments
they had, you know, it was different for each

person. Well, 90% of that variation is explained
by the equation. And of course, if you take

100 minus 90%, there's 10%, unexplained variation.
So there's still some variation that could

be explained by other variables, but not a
lot. And how you actually stated is, you know,

when you're done with this, if you were writing
a paper, you'd say, 90% of the variation in

the number of appointments is explained by
DBP. And I know people are like, explain,

like, it doesn't have a mouth, like, what
does it talking about? You just have to say

it this way. There's it's statistics ease,
this is how you say it. And by

contrast, or by complimentary, what you would
say is 10% of the variation in the number

of appointments is not explained by DBP. Right?
It could be explained by other things. Well,

we happen to get a nice, I see CD for coefficient
of determination. You know, we got a nice

high one. But what if it's a low? Well, let's
just think about it CD should be better than

at least 50%? Because that would be random,
right? And the higher the better. So if you're

on a test, nobody's going to give you a CD
of like 60% and say, Is this any good because

I don't know, you'd be very conflicted. In
real life, what I use it for is to compare

models, if one is 60%, and the others 55%.
Of course, I'm going to go with a 60%. One,

but it's still not very good, right. And if
it's low, you know, the higher the better,

basically. And if it's low, it means that
you probably need other variables to help

the x you use to explain more of the variation
because that x is not doing. Okay, in summary,

I just wanted to go over chapter four, so
you realize where we've been. Okay. So we

started out with a set of quantitative x,
y pairs. First thing we did was we made a

scatterplot, we wanted to look at the linear
relationship between x and y. And we wanted

to look at outliers. If we'd seen a lot of
outliers, or no linear relationship, we would

have stopped there. But because this is a
class we had to learn, I forced them to be

a scatterplot with a linear variation, and
not too many outliers. So we could move forward

and do our so we calculated our to see if
our correlation was positive or negative,

and weak, moderate, or strong. So that's what
you do if you find a linear relationship.

Next, in addition, in this lecture, we calculated
B and A to come up with the least squares

line equation. And I just wanted to you to
notice that the sign on B will always match

the sign on R. So if you have a positive R,
you'll have a positive slope, if you have

a negative or you have a negative slope, but
otherwise, the numbers won't match, just a

sign. And then also, I wanted you to notice
that strong correlations will give you high

coefficient of determination, even if they're
negative correlations, because remember, it's

r times r. And so negative times negative
are still as positive, right? So if you have

strong correlation, like negative point nine,
or point nine, it really doesn't matter what

direction if it's strong, then you're going
to get a high coefficient of determination.

So after we did this B and A thing, we use
that linear equation to calculate residuals,

right, like we took the x's from the original
data and put them in got the Y hat and calculated

the residuals. After that, we use R to calculate
the coefficient of determination or CD, to

decide if we wanted to use the literate equation
for prediction. Because if it was bad, we

weren't going to do that. But we decided was
good for prediction at 90%. And we decided

to use it. So that was our journey through
these xy pairs all the way down to the coefficient

of determination. Good job, you made it. So
in conclusion, the least squares criterion,

and calculating the least squares line was
the first thing we went over how to do that

and what it all means. And then I reviewed
some issues with prediction using the least

squares line, because it looks kind of easy.
It looks kind of, you know, better than sliced

bread, but there are some things you have
to think about. Finally, we went over the

coefficient of determination so that you could
figure out how good your least squares line

equation was. And I just wanted to point out
that CD kind of looks like CDs, you know,

like we used to have CDs. They were so pretty
and rainbowy like that. But now all CD means

is coefficient of determination. Hello, and
welcome back to statistics. It's Monica wahi

are labarre College lecturer and You've made
it to chapter seven, I broke up chapter seven

into bite sized pieces. And we're going to
start with chapter 7.1, talking about the

normal distribution and the empirical rule.
So here are your learning objectives for this

lecture. At the end of this lecture, you should
be able to state two properties of the normal

curve, state two differences between Chebyshev
intervals and the empirical rule, and explain

how to apply the empirical rule to a normal
distribution.

So, remember, distributions, we learned about
them a while back, but I'll remind you a little

bit about them. And then we're going to talk
about properties of the normal distribution,

or specifically the normal curve, that shape
that comes out of making a histogram of normally

distributed data, then we're going to remember
Chevy Chevy intervals, we're going to talk

about what Chevy Chevy did for us, and what
Chevy Chevy really didn't do for us. And then

we're gonna move on to the empirical rule,
which works very well, better than Chevy Chevy

intervals, when you have normally distributed
data. And then I'm going to show you an example

of how to apply the empirical rule to that
normally distributed data. So remember, the

normal distribution, in fact, remember distributions
at all right? So to get a distribution, and

a lot of people sort of forget this, by the
time we get to chapter seven, but I just wanted

to remind you, this is from an earlier lecture,
we had a quantitative variable, which was

how far a patient's had been transported.
And we determined classes, and we made a frequency

table. So remember that. And then after that,
we made a frequency histogram, and then made

a shape. And as you could see that shape,
which is the distribution, that shape in this

one was skewed, right, see that light on the
right, okay, but that's an example of something

we cannot apply the empirical rule to, because
the empirical rule only applies to normally

distributed data. So I had to give you an
example of that. And here's my example. So

when I was in my undergraduate in costume
design at the University of Minnesota, they

made us take a chemistry class and one of
those big lecture halls. So I was in a very

large class that probably had about 100 people.
And we were given this really difficult test,

it was 100 point test, and I was used to getting
like A's. And so when they were done with

the test, the T A's, were handing the tests
back to everybody. So they could see their

grade, while the professor was writing on
the board, and was reading the frequency of

all the different scores. And I remember the
TA handed me my test, and it said 73 on it.

And I'm used to getting like 90s, up to 100.
And I remember stating out loud, saying 73,

that is an awful score, I can't believe I
did so badly. I was talking like that. But

at the same time, the professor was writing
the frequencies on the board. And what I realized

is the top score was in the 80s. And I had
the third top score was 73. That's how hard

the test was. And that's a nice Shut up, because
I noticed everybody giving me dirty looks

because they had scored actually below me.
So I wanted you to imagine that class. And

I imagined what the normal distribution would
look like for that class with the distribution

of the scores. And the reason why I thought
it would be normal is because we all did badly,

right. And so nobody got 100. So we were all
below the 100. So I imagined this curve here

for you. And I imagined my class, I had 100
people just to make it easy. Of course, the

test was difficult. And nobody got 100 points.
And the mode, the median. And the mean, were

all near see great, because you remember how,
when you have a normal distribution, the mode,

median, and mean are all on top of each other.
So we all did pretty badly. So I'm going to

use this example of the fake chemistry test
scores to exhibit exemplify these properties

of the normal curve. So there's five I'm going
to talk about. The first is that the curve

is bell shaped with the highest point over
the mean. And so you can see I drew a scribbly

little curve, put a little arrow there to
show you that that's where the mean of the

scores were. And then I also wanted you to
notice that the curve is symmetrical with

a vertical line through the mean. So there's
like a mirror image of the curve on either

side. Now, it's not perfect, obviously. But
it should be roughly like that. And you know,

this is not true of skewed or bi modal or
these other things we've been talking about.

Okay, and the third property is that the curve
approaches the horizontal axis but never touches

it. You don't have to memorize this, but remember,
asym totw or asymptomatically close, that's

when a line gets really close to another line,
but they never touch.

It's so romantic. But anyway, that's a very
Bollywood thing to say, by the way, but uh,

so the curve approaches the horizontal axis
and never touches or crosses and then also

there's this inflection or these transition
points between cupping upward and downward.

And these transition points occur at about
the mean, plus one standard deviation and

about the mean minus one standard deviation.
And this is a little hard to explain. But

imagine you're on a roller coaster and you're
going up this normal curve. There's this part

where you're just mainly going on, well, the
part where it seems to kind of level out and

you're at the top of the curve, he starts
to relaxing. That's that inflection point.

And so as you're going over in the roller
coaster, and you're in that flat part, and

then you start kind of going down, that's
the second inflection. So that's where what

it's saying about is the property of this
curve is that you have these inflection points

like that. And they roughly occur at plus
or minus one standard deviation above and

below the mean. Then finally, and I call it
this, and just so you could see it, the area

under the entire curve is one, so think 100%.
So it would be nice if that were a square

or rectangle, or even a triangle, something
that we're used to in geometry, but it's not,

it's this goofy shape, right? But still, you
need to get it in your head that that shape

is worth 1.0 in proportion land, or 100% in
percent land. And what I mean by that is,

let's say we cut that shape and half, the,
each side would have 50% or point five on

it, then let's cut it a different way. So
the part of the curve on the right side of

that line is a fourth of the curve, or 25%
of the curve, even though it's goofy shaped,

and the part on the left side is 75%. So that's
what we're trying to get you to think like

is that, yeah, you can just declare that all
the area under the curve equals one or 100%.

But the reason why we're declaring that is
because we're gonna cut it up and say different

amounts of percent of the curve. Now we get
to the empirical rule, since we reviewed this

whole curve thing, and I'm going to make you
remember Chevy shove, I'm sorry, but you know,

let's talk about Chevy Chevy, Chevy shove
helped us get some intervals, right, in intervals

have boundaries, or limits, they have a lower
limit and an upper limit. That's how you know

what bounds the interval. So when we were
doing Chebyshev intervals, what we would do

is we'd figure out a lower limit and upper
limit, and we'd say at least so much percent

of the data falls in the interval, right?
So when we would choose the lower limit of

mu minus two times the standard deviation,
and the upper limit was mu plus two times

the standard deviation, we would say at least
75% of the data were in the interval. So I

wanted to just show you a demonstration using
my fake class. So remember, there were 100

students in the class, I actually came up
with a mu for them. And their mu on the test

was 65.5. So my 73 was better than the mean,
but not much better, right. So the mu for

that class was 65.5. And the standard deviation
was 14.5. So I calculated these chubby shove

this championship interval for 75% of the
data. So I took 65.5 minus two times 14.5.

And I got 36.5, which is a pretty bad grade.
And then the upper limit was pretty good,

right? 65.5 plus two times 14.5 equals 94.5.
On 100 point test, that's a pretty good grade,

right? So if you had 100 data points, or 100
students, at least 75 would have scored between

36.5 and 94.5. So you're probably already
realizing, okay, that doesn't really help

Monica, who scored 73. And this is a really
wide range, we say at least 75% of people

score there, you could probably guess that
without even knowing about chubby ship intervals,

right? So it didn't really help me narrow
down, like how well is this class doing? If

I had had the mu and the standard deviation,
I could have calculated this and said, Okay,

I'm no better off.

So championships theorem on the left side,
and applies to any distribution, you don't

need a normal distribution, you can use that
skewed distribution. Also, you'll notice it

says at least. So like this was at least 75%
of the data fell in there. Maybe even 100%

fell in there. So it doesn't really help us.
And as you go, let you start with two standard

deviations. If you go out three, it's 88.9%.
And four, it's 93.8%. You know, you might

as well start at the beginning and say almost
100% of the data falls in this interval. And

if you're saying that it's not very useful,
right. But it kind of gets stuck doing that

because championships theorem applies to any
distribution, the empirical rule is much more

elite. It only applies to the normal distribution.
And you'll see why if you are lucky enough

to get the normal distribution that you want
to use the empirical rule instead of championship.

Okay? Because Secondly, the empirical rule
says approximately It doesn't say at least,

so it's saying basically, not at least it's
saying about exactly this. So you can trust

it. Okay, you don't have like this unknown,
like maybe 100%. There's, so it says, This

is what it says and I'll show you a diagram
of it, but it says that 68% of the data are

in the interview interval. mu plus or minus
one standard deviation. So mu minus one standard

deviation all the way up to mu plus one standard
deviation 68% of the data are in there. And

you'll notice that Chevy chef didn't even
say anything about one standard deviation.

And so already, we've got something way more
useful if we apply the empirical rule, right.

So next we go to 95% of the data are in the
interval, mu plus or minus two standard deviations,

95%, approximately 95% are in there. Now,
if we had bought chubby chef, we'd be saying

about this too, we'd be saying 75%. Okay,
we'd be saying at least 75%, which could be

95%.

But here, if we're using the empirical rule,
we're relatively sure that it's 95% between

mu plus or minus two standard deviations you
can like better, right? Finally, if you get

out to three standard deviations, you're kind
of running out of data, because 99.7%, almost

all of them fall in that interval. So as you
can see, the empirical rule is going to give

you a more specific answer. But again, you
can only use it if you have a normal distribution,

but which we do. So let's go look at that.
Okay, this is a diagram that I'm going to

help I made it myself, actually, because I
thought it was the other diagrams I saw were

not pretty. And this one is very pretty in
my mind, but let me unpack this diagram for

you, because there's

a lot going on. And

first of all, I want you to notice the shape
of it, it's a normal distribution, okay. And

then I want you to notice that I put this
black line down the middle, and I put a little

arrow that says mu. So this is where we want
to imagine mu, it's no matter what your what

your actual numbers are from you. Like in
our case, this is 65.5 for our points. Just

imagine whatever your mu is, and whatever
your standard deviation is, this is where

you would put the meal, right, then you'll
notice that each of these sections that's

colored, has a little standard deviation symbol
in it, because that's representing that, that

the width of that is one standard deviation.
So if your standard deviation was like five,

then mu would be plus plus or minus five,
like the green one would be mu plus one standard

deviation. So it'd be mean plus five, and
then you draw that parallel line there and

see that arrow that says mu plus one zero
deviation, that would be there. And of course

I can, I just had to use the symbols, because
I don't know how big the standard deviation

really would be, or what the mean really would
be. But whatever it was mu plus one standard

deviation, if you go up there, you would see
that that green area represents 34% of the

data. And if you're lucky enough to have exactly
100 people, like I did in my demonstration,

that would mean that between mu and mu plus
one standard deviation of these test scores

would be 34 people's scores, right, so you
can really figure that out. Same with the

yellow section only, that's mu minus one standard
deviation, and 34% of the scores would be

between those two

numbers.

Now you'll see as you get up into the blue,
that's between one and two standard deviations

above the mu, you'll see that because the
roller coasters a lot lower to the ground

there, that section is really small, it's
only 13.5% of the data. And the same with

the orange one that's on the other side of
the mu. So that's below the mean. And that's

only 13.5. And then you'll notice that at
three standard deviations, between two and

three, there's a little tiny piece right,
the purple piece and the red piece, those

are only worth 2.35% of this shape. And then
I wanted to point out there is some stuff

at the end, in the little black part beyond
three standard deviations on either side,

there's point one 5%. And a lot of times people
forget that. But one way you can make sure

that you've got to remember that it's there
is that if you add up all these percents on

the slide, you'll get 100% because remember,
I promised you that the whole the whole curve

is worth 100%. And this is how we split it
up. I also want you to notice that there's

kind of a cheat, right? If you just add up
the green, blue, purple, and then the little

black part at the end, if you just add up
those percents, you'll get 50%, right, because

that's half the curve. And the same, you'll
get the same thing if you do the yellow, orange,

red, and the little part and the black at
the bottom. If you add those up, you'll get

50%. So that's how you want to just conceptualize
this whole empirical roll diagram. But now

we'll apply. So I put the empirical rule diagram
on the left, and then I put our class frequency

histogram on the right and look, I put the
meal and I put the standard deviation so we

could have it there. Now the first part of
this section, I'm just going to show you how

to fill in the numbers under the diagram.
Okay, and then after we fill in the numbers,

I'm going to talk to you about how to interpret

those numbers.

So let's start with easy let's write the mu
underneath the symbol for me, which was 65.5.

So we just wrote that was simple, okay. Now
let's do the plus or minus one standard deviation.

So you'll see 65.5, which is our mu minus,
and I put one times 14.5. I know I just did

that for demonstration purpose. So you see,
we're doing one times the standard deviation.

So if you subtract that from the meal, you
get 51. And so I wrote that 51 underneath

the mu minus one standard deviation. And if
you go the opposite way, and you add on 14.5,

you get 80. So I put that up there. So that's
I just labeled those two, you can kind of

guess what we're going to do on the next

slide.

Surprise, we're going to do almost the same
thing. All we're doing the mu minus two times

the standard deviation to get the 36.5. And
the mu plus two times the standard deviation

to get that 94.5. And you probably already,
we're ahead of me with this one. This is where

we do 65.5 minus three standard deviations,
and we get 22. And then we add three standard

deviations, and we get 109. And now we're
all able to So what does this all mean? Well,

remember, our n equals 100, just out of convenience.
So what does this mean? It means that 34%

of the scores are between 51 and 65.5. So
that's the yellow bar. Right? So 34 scores

were that because I 100 people in the class.
So I'm standing there in that class, and I've

got a 73. But I don't 34 of those people I'm
looking at have a score between 51 and 65.5.

I also know that another 34%, or another 34
in this class, because there's 100 have a

score between 65.5 and 80. And my 73 is somewhere
in there, right? So already, I'm getting an

idea that 68 people are 68% of the scores
are going to be between 51 and 80. Right.

And so I'm right there with 68% of the class.
So I'm going to go through some fake test

questions for you to just show you how to
come up with the answer. So let's say the

question was, what percent of the data student
scores are between 36.5 and 80? So think about

how you would answer that question. So see
where 36.5 is, it's on the lower limit of

the orange part, and see where the ad is,
it's on the upper limit of the green part.

So what you would do is you would add up the
percents in between right 13.5 plus 34, plus

34? And the answer to what percent of the
data are between 36.5 and 80? The answer would

be at 1.5%. Here's another question. What
cut point marks the top 16% of the scores.

So already, you know you're up in that area,
probably where the purple or the blue are,

right? And so what would make the top 16%?
Well, if you actually add together that point,

one 5%, from the little black part, the 2.35%,
from the purple, and the blue 13.5%, you'll

get 16%. So the cut point then for that all
the scores above 80, that would constitute

the top 16%

of the scores.

Here's another quiz question, what percent
of the scores are below 94.5. So we see 94.5

is at the upper limit of the blue section.
So you could kind of say, well, let's just

add up everything below. Right, we'll add
up everything below it, and that person, the

scores will be below 94.5. And so we do that
we add everything below it. But remember how

I said that there that the yellow, orange,
red, and the little black part there that

that equals 50%? If you just wanted to say
okay, that's 50% plus the green part, plus

the blue part, you could do that, and then
you get the same answer. So what are the cut

points from the middle 68% of the data? I
just wanted to show you an example. What if

they say middle, right? Well, you're gonna
have to be centered around me that right?

So the middle 68% means 34% above the mean,
and 34% below the mean. So the cut points

would be 51 to 80. Okay, now I'm going to
ask a similar question, but I'm going to use

different words. Okay. What is the probability
that if I select one student from this class,

that student will have a score less than 80?
Okay, so notice, I'm using totally different

terminology. I'm saying what is the probability
yet? The only the actual answer is what you

would probably guess, which is where you add
up all the percents below 80. So the point

of me giving you this quiz questions is to
point out that percent and probability mean

the same thing when you talk. So either I'm
gonna say what percent of the data are below

at the score of 80? Or what is the probability
that if I select one student, that student

was scored less than 80? That is actually
the same question. So the answer is going

to be I use that 50% trick here. That answers
me 50%, which is the whole bottom half of

that curve plus 34% gets up to 84%. Right?
So, so the probability that if I select on

student, that student will have a score less
than 80 is 84%. And that's the same as what

percent of the data is below 80 is 84%. Okay.
Here's another probability question, what

is the probability I will select a student
with a score between 36.5 and 51? Well, that's

as if I was asking, it's the same question
as what percent of the data are between 36.5

and 51? which you would know the answer that
that would be 13.5. That's the orange part,

right? But even if I say, what is the probability,
I will select a student with a score between

36.5 and 51 13.5%? So let's say that we were
at a casino, and we were betting, right. And

I'm like saying, okay, there's 100 students,
I'm going to just grab a score out, and I'm

betting a lot of money that I'm going to grab
somebody between 36.5 and 51. And you'd probably

be like, you don't want to bet on that. Because
you only have 13.5% probability of selecting

one, you probably want to bet if you're going
to bet on something in the in the yellow section

or something in the green section, because
they have higher probability. So that's how

you would think about probability. And percent,
even though they're kind of the same thing.

I just wanted to show you how they word the
questions differently.

But it means the same thing. So now I want
you to just sit back and think for a second.

So think about what would happen in a different
class taking the same hard test, meaning nobody's

getting 100%? What's the mu was the same,
meaning everybody's doing badly. But the standard

deviation was larger than 14.5? What would
that do to the intervals? So let's just stare

at this for a second. Let's say the mu was
still 65.5. But the standard deviation was

like 30. Okay, there was a lot of variation
in the class, that would already mean that

where the ad is right now, that that would
actually be 95.5. Right? And where that 51

is there. Now, if we have a standard deviation
of 30, that would actually be 35.5. I mean,

that'd be a way bigger interval, right. And
so the class I was in in chemistry was an

undergraduate class, I was in costume design.
This was a whole bunch of different kinds

of people in chemistry. And that's probably
why we even had kind of a big standard deviation

of 14.5. Even though I made that up. I mean,
in reality, we probably did have a big standard

deviation. I knew in the chemical engineering
department, they had chemistry classes for

chemical engineering majors, I'll tell you,
their standard deviation was probably a lot

smaller, because they were probably more alike
and got more similar grades as each other.

But with this diverse class, we probably had
a pretty big standard deviation. So that gets

to my last question, what if the standard
deviation was actually smaller than 14.5.

So if we were like in the chemical engineering
class, and they were taking chemistry, and

they had a smaller standard deviation, maybe
they might have had the same mean 65.5. But

let's say their standard deviation was like
five, then where the ad is now would be a

70.5. And where the 51 is, would be a 60.5.
And we'd have way more confidence of where

we knew the scores fell, like as I was standing
there with my 73. I would be saying like,

Oh, you know, my 73 is pretty high, if everybody
has a small standard deviation, right? Whereas

it's not that high here, because we have kind
of a big standard deviation. That's in the

first though the green part. So the reason
why I want you to think about that is, that's

why

this

shape goes by mu and standard deviation, because
it really matters how big the standard deviation

is, how big each of those areas are with the
different colors. So I just wanted to remind

you that percent, area and probability are
all related. The percents literally refer

to the percent of the area of the shape, okay?
And imagine the whole thing is 100%. So just

to remind you, the orange part is 13.5% of
the area of the hole shape, but it also is

the probability that an X like a student and
x falls between mu minus one standard deviations

and mu minus two standard deviations. And
that if I select 1x, from a group, this group

that I'm 13.5% is the probability that I will
get an X in that range. And so it means both

things. So in conclusion, the empirical rule
helps establish intervals that apply to normally

distributed data. And it's more useful than
trebuchet. Because it's more specific, these

intervals have a certain percentage of the
data points in them. And they also refer to

the probability of selecting an X in that
interval. And these intervals depend on the

mean and the standard deviation of the data
distribution. So if those change then exactly

where the numbers are on those intervals change.
Well, I hope you enjoyed my explanation of

the empirical rule. And now you can practice
doing it yourself at home. Good morning, good

day. And good afternoon. This is Monica wahi,
your library college lecturer here moving

you through chapter 7.2, and 7.3, z scores
and probabilities, I decided to merge these

two chapters together, because I thought they
actually kind of belong together, I didn't

really understand why they were separated.
So at the end of this lecture, you should

be able to explain how to convert an X to
a z score, show how to look up a z score in

a Z table. Explain how to find the probability
of an X falling between two values on a normal

distribution, describe how to use the Z table
to look up a z corresponding to a percentage,

and describe how to use the formula to calculate
x from a z score. Well, that sounds like a

lot, but you'll understand that at the end
of this lecture, first, I'm going to go over

what a z score is and what the standard normal
distribution is. Then I'm going to talk about

Z score probabilities. And what those are,
I'm going to show you how to use the Z table

to answer some harder questions besides the
ones I talked about during the z score probabilities

section, then I'm going to show you how to
use a slightly different formula to calculate

x from z. Finally, I'm going to just remind
you some tips and tricks about using z scores

and probabilities correctly. So all this talk
about z scores. So what is the z score? And

what is the standard normal distribution?
Well, let's take a look at this very, pretty

thing I made. You may recognize it from the
last lecture, it was my little Empirical Rule

diagram. So remember, the empirical rule,
remember how it required a normal distribution?

Well, that worked well for the cut points
available, right? Like mu mu plus or minus

one standard deviation, mu plus or minus two
standard deviations. If we ask questions that

were right on those cut points, we had good
answers. But what about in between those cut

points. So I wanted you to notice, in this
Empirical Rule diagram, these numbers at the

bottom, like I just circled them, like negative
three, negative two, negative one, and then

mew doesn't have a number. So pretend there's
a zero there. And then there's one, two and

three, okay? That is the standard normal distribution.
And that is also called z. So these things

on the right, those are z scores. So see the
green area, zero is the z score that's on

the lower limit of that, and one is the z
score at the upper limit of the green area.

So you can see that this whole curve, the
the standard normal distribution on the right,

the whole, the mean of the whole curve is
zero. And the standard deviation of the whole

curve is one. And that is what c score is.
So I just want you to notice the concept of

standard. I'm, I'm in the US. And in the US,
we use, you know, the US dollar, but one of

the things I've noticed is that a lot of countries
see it as a standard. So they'll map their

currency to the US dollar. So maybe the Euro
will map its currency to the US dollar, maybe

the Egyptian pound

will also map its currency to the US dollar.
And once it does that, it's a lot easier to

compare them, right. And so that's the main
reason for the standard normal distribution

is it helps you compare exes from different
distributions, different normal distributions

that have different means in different standard
deviations from each other. It helps you map

them to this normal standard normal distribution
here that standard, so you can compare them.

So let's talk about z scores, every value
on a normal distribution. So every x can be

converted to a z score, just like I was saying
how you can convert any currency to dollars,

there's some

formula for that.

You can convert every x on a normal distribution
to a z score. But you have to know how to

use the formula right? And what goes into
that formula. Well, first, you need the X

that you want to convert to a z score. So
you need to pick one, then you need to know

the mu of your distribution, your normal distribution,
and the standard deviation of your distribution.

And here are the two formulas that are used.
The one I was just talking about is on the

left is the formula for calculating the z
score. And we'll go over the one on the right

later in this lecture. So remember in the
last lecture, I was talking about a class

that had 100 people in it. And that all took
a really hard test, it was so hard, nobody

got 100%. And it was 100 point test. So nobody
got 100. The top score was in the 90s. So

um,

and remember, in the upper right there was
there's the meal, the meal was 65.5, which

is pretty bad score, 100

point test, and the standard deviation was
14.5. So I'm going to give you an example

of calculating a z score on that particular
distribution. So let's say you got a friend,

you have smart friend, and that's my friend
got a 90 in the face of all this? Well, let's

calculate the z score for 90 on this particular
distribution. Okay, so here's what we're going

to do is, first we're going to remind ourselves,
you don't have to do this in real life when

you're doing it. But I'm just doing this for
demonstration purposes, is what our Empirical

Rule stuff look like. Remember, at mu plus
one standard deviation was 80. And mu plus

two standard deviations was 94.5. So already,
you know, whatever your answer is going to

be for 90 is it's going to be between one
and two. Right. But we just don't know exactly

what it's going to be. So I'm just showing
you this for demonstration purposes to relate

it to the last lecture. But you don't have
to do this in real life when you calculate.

Okay, so we know that the Z we're going to
calculate is going to be somewhere between

one and two. And as you'll see, on the slide
here, I labeled over on the z curve, I labeled

where z equals zero, which is the mu that's
65.5. So we're going to anticipate we're going

to get a z score, that's somewhere between
one and two. And you'll see in blue, I listed

the ingredients, right, so we have the smartphone
score 90, we have the mu 65.5. And we have

standard deviation 14.5. And then we have
our z formula. So let's do it. Okay, so x

minus mu is going to be 90, which is our x
minus 65.5. You do that out first, and then

you divide it by 14.5. And look, our Z score
is 1.69. And that's exactly where we thought

it would be, it would be somewhere between
one and two. And so as you can see, you can

take any x and convert it to Z. Here we'll
do another example, only this friend is not

so smart. This friend actually got a score
that was kind of low, it was so low, it was

below the meal of 65.5, this poor friend only
got a 50. So let's try it again, let's do

a z score for 50. So again, you know this
is just for demonstration purposes. But remember,

in Empirical Rule land 51 was that mu minus
one standard deviation. So we're going to

expect that between again, negative one and
negative two is z is where our 50x is going

to land if we calculate the z score. And so
here we are, we calculate the z score, we

have 50 minus 65.5 divided by 14.5, and we
get negative 1.07. And the reason why it's

negative is, as you can see, it's on the left
of the meal,

so then the z score is gonna be negative.
And so as you can see, it's exactly where

we thought it would be, it would be a little
bit to the left of negative one.

So now we're going to get into something that's
a little bit harder, which is the z score

probability. So you're feeling pretty good
about the z score. But now let's talk about

the probabilities. Okay, so remember the probability
from the empirical rule, this is just old

Empirical Rule stuff. So remember, I gave
you a question at the end of that lecture,

I said, What is the probability I will select
a student with a score between 36.5 and 51?

And remember, the answer was like this orange
area, which is 13.5%. But what if you have

z scores like 1.69? The Smart friend, and
negative 1.07, which are the not so smart

friend, you know, in other words, you have
excess of 90 and 50, which are not on the

empirical rule? How do you figure out the
percent or the probability? That's the next

step with your z scores? Okay, so now let's
ask this question, let's say, what is the

probability that students scored above the
smartframe. Now, we could also ask for below,

but I'm just choosing to ask for above this
time. So in other words, what is the area

under the curve from z equals 1.69? All the
way up. So see, like a little ways through

that blue edge. We wish we knew the area for
everything up from 1.69 Z, through the purple

area through the little black thing at the
top. We wish we knew that area. We only know

from the empirical rule what's on the cut
points of like one and two, but we don't know

this in in between things. So how do we figure
that out? Well This is another problem here.

What is the probability that students scored
below the nozzle smart friend, right? And

in that case, see the diagram, we'd have to
figure out what is the part of the orange

that that friend gets plus the red and plus
a little black part of the bottom? What is

the percent or the proportion of the curve
that represents that. So that's what we're

getting into now. And that's what we do is
we look these up in a Z table. So what the

Z table is, is basically, they figured out
every single Z score, you could have between

negative 3.49. And I'll go into why negative
3.49, between negative 3.49 and positive 3.49.

And they went like every 100.

So

they figured out for every single one of those
these scores, what the probability is, and

they actually fit that all on a table. And
so now, what I'm going to show you how to

do is how to use that table to look up the
probabilities. And by the way, if you look

up a probability that happens to be on one
of those Empirical Rule cut points, you'll

get what the empirical rule says. It's just
said, the empirical rule is nice, because

you don't have to pull out the table. But
if you have something that's not on the empirical

rule, cut points, get out your Z table. So
how do you use the Z table? Well, the first

thing is you want to figure out what area
you want, right? So we're going to start and

do the not so smart friend, because that's
a little bit easier actually to demonstrate.

Okay, so what is the probability that students
scored below the not so smart friend? So,

which is a secret way of saying, what is the
area under the curve that makes up most of

that orange part, all the red and the little
black part at the bottom? What is that proportion.

And so for areas left of specified Z value,
you're supposed to use the table directly.

So I'm going to show you how to use that table
to look up negative 1.07. And then I'm going

to come back and tell you what they mean by
use it directly. Hi, there. So here we are

at the Z table. And if you have the book,
you can look it up in the appendix in on page

eight. But there's also a lot of z tables
on the internet. Sometimes they're arranged

a little differently. So I'm using this one
because it's from the book. So remember, the

Z that we're looking up, we're looking up
the Z of negative 1.07. So remember, I said

they had to somehow calculate all the different
probabilities for every single z between negative

3.49 through positive 3.49. Every 100th, they
had to come up with that, well, how did they

fit it all on their table? Well, this is what
they did. See, this is the being the Z table.

Remember, I said negative 3.49? Well, this
is negative 3.4. And then to find the Z and

negative 3.49, you have to imagine that the
nine is here, but it's going to be the last

one here. So see this nine here, this is what
it would be. So just for pretend, if we had

a z score of negative 2.58, I go 2.5. And
then I have to go over to the eight, one right

here. Okay. Or if I had one that was negative
2.10, right, or negative, just plain 2.1.

Right? Then I'd go over just one to this zero,
line and see these these little tiny things

in here. Those are all probabilities. In fact,
let's go look up our probability, which is

negative 1.07. So we're going to go down here,
negative, here we are at negative 1.0. And

then we have to go over to the seven column,
right, so what's the song? Here's a song,

it's three from the left, I guess I could
have guessed that. So we have negative 1.0987.

So this is point 1423. Otherwise known as
14 point 23%. So that's actually what you

get out of the Z table. That's the probability
that's the percent you're looking for. And

just in case, you're wondering, these aren't
all negative, the first page is negative.

The second page is positive is all the positive
Z scores all the way up to 3.49. But what

I want you to hold in your head is what we
just looked at, which was negative 1.07, which

is point 1423.

Okay, hold that thought.

Okay, here we are back at our slides. And
so look at that green part where it says four

areas to the left of a specified Z value,
which we're doing with the not so smart friend,

use the table entry directly. So here was
our table entry. It was point 1423. So we're

just going to use that number that we found
and we're gonna say the probability then,

is 14.23%. And that kind of makes logical
sense knowing the empirical rule. Now, I'm

going to show you an example of what why I
was saying, use it directly. In this next

example, we're going to look at the smart
friends probability. In fact, we're going

to ask what is the probability that the students
scored above the smart friend in the smart

friend set z equals 1.69. So I'm going to
demonstrate now, for areas to the right of

a specified Z value, you either look them
up in the table, then subtract result from

one, or you use the opposite z, which is in
this case would be negative 1.69. And you'll

get the same answer, whether you do with the
first way The second way, but I'm going to

demonstrate both okay. So first, I'm going
to demonstrate what happens when you look

up the probability in the table for that,
see, and then you subtract that probability

from one. So let's go look up z equals 1.69.
All right, here we are back at our Z table,

only this time, we're looking up a positive
z. So we don't want this first one, we want

the second one. So remember, we're looking
up z equals 1.69. So we're looking under here

for 1.6. And that's right here. And now we
have to go over to the nine column. So that's

going to be point 9545. So hold that thought,
point 9545. Okay, we're back with our probability

that we looked up in the Z table. Now remember,
we were supposed to look it up in the table

and subtract the result from one. So that's
what we're going to do now. So we found point

9545 in the table, we're going to take one
minus point 9545. And we get 0.0455, or 4.55%,

this little tiny piece, which kind of makes
sense, because it's right at the top of the

distribution, just a little piece of the blue,
and the purple, and then the little black

at the top. Alright, and so what you want
to imagine is that point 954, or five, which

is like 95.4, or 5%, that's the whole piece
below z equals 1.69. That's most of the blue,

the green, the yellow, the orange, the red,
and the little black at the bottom, that's

all in the point 9545. Okay, so again, we
were looking up in the area to the right of

the specified Z value, and I showed you the
first way of doing it, there's another way

of doing it, and that's where you just use
the opposite z from the get go. So we're going

to now use the opposite seat, we're going
to look up negative 1.69. All right, here

we are back at the Z table. Only this time,
we're looking at negative 1.69. So negative

1.6 is the first thing we need to find in
this column. So here we are negative 1.6.

And then we know nine is the last column.
I'm learning that. So we'll go over here.

And so that that looks familiar. Right point.
Oh, 455. Okay, hold that thought. All right,

well, back. And so as you know, if you look
it up in the table directly, like the 1.69

directly, and you take that probability, and
you subtract it from one, which is what we

did last, we got the same answer we got now,
right point, oh, 455, or 4.55%. So it is kind

of more efficient, to just use the opposite
z, if you're looking for areas to the right

of the specified Z value. But I always say
when you're done looking it up, compare it

to the picture. And I always say draw a picture
to, you know, I don't mind if you have normal

curves drawn, drawn over all of your homework,
or all over the wall, I guess, or maybe a

whiteboard, that's probably more efficient.
But it's best to draw it out. label on there,

where your z and your x are, and then just
look at it. Because we know that the little

piece above z equals 1.69 is not 95% of that
curve. It's just not it, that's over 50%.

And we can tell that little tiny pieces under
50%. So if you accidentally do the first way

and forget to subtract from one, you know,
maybe if you check it against your normal

curve drawing,

you'll realize oh, I made a mistake. So even
though there's two different ways to find

the probability, if it's to the right of the
z value, just try to make sure no matter which

ways you use that you finally do a reality
check against the drawing you make, just to

make sure you got the right piece because
there's only two pieces. There's a big piece

and a little piece of the skirt, and we got
4.55% we know that's a little piece and we

know From our drawing that we were looking
for the little piece. So that's how you do

your reality check. Okay, you thought that
there weren't any harder questions? Well,

here are some harder questions. So this is
a little bit more on probabilities in the

Z table. So here's another question we haven't
handled yet. What if you were looking at a

probability between two scores, such as the
probability the students will score between

50 and 90, so it's somewhere in the middle,

okay.

Note that in that case, when you have a between
one, you actually have two axes, and we'll

label them x one and x two, so the not so
smart friend is going to be x one, and the

smarter friend is going to be x two, just
to keep these x's straight. Okay. So the next

step is you're going to calculate z one and
z two. And I'm kind of cheating. Because we

already did these, we already knew the Z one
for the National smartphone was negative 1.07.

And we already knew the Z two, for the smarter
friend was 1.69. So I just put them on the

diagram. Okay, and then here's this beginning
of the strategy, and I'll just explain the

strategy, and then I'll do the strategy. So
for z one, you find the probability to the

left of the Z, so you find the little piece
to the left. And remember, you can take the

direct probability from the Z table. So that's
what direct means is you just get to copy

it directly out of this table. Then for z
two, you find the probability to the right

or above z. So you find the little piece there.
And you use one of those two methods I showed

you, which we did together. And then finally,
imagine like the whole curve, you're subtracting

the piece at the bottom, the Z, one probability,
and you're subtracting the piece at the top.

So you're trimming with those two pieces to
get the between probability. So that's the

strategy is basically you find out the the
size, the probability of each of the little

pieces on the sides, you subtract both of
those from one, and that traps whatever's

left in the middle. So I'll demonstrate this.
So remember, for z one, the probability to

the left of Z one was point 1423. We did that
together. And then we use both of those methods.

And they got the same answer to find the probability
to the right of z two, which was point o 455.

Okay, so that's a little piece at the top,
and then we got the little piece at the bottom.

And now we'll take one minus the piece at
the bottom minus the piece of the top and

the total is point 8122, or 81. Point 22%.
which kind of makes sense, that's a big piece

in the middle. So it wouldn't be surprising
if it was about 80% of the curve. So this

is how you do a between like. Here's another
question I haven't really handled, what have

you looking at a probability more than 50%?
So such as the probability that students will

score greater than 50? Right? Like, like the
big side? Okay? Well, actually, you just do

what you normally would do, you say four areas
to the right of the specified Z value, either

look up in the table and subtract the result
from one, or use the opposite z, which in

this case would be 1.07. So if we did method
one, we'd end up going one minus point 1423,

which we already looked at, and we get point
8577, we use method to we'd take the Z of

1.7, not negative 1.07, but 1.07. And we could
go look it up in the Z table, and we get point

8577. Again, 85 point 77%. So if this isn't
actually a harder question, I just wanted

to show you how it works when you're getting
like a bigger piece, bigger than 50% piece

of the distribution. And here's another sort
of similar example, where we're looking at

the probability that students will score less
than 90, okay. So that's easy, right for the

area's to the left of the specified Z value,
just use the table directly. So when we went

and looked up z equals 1.69, we got point
9545. So that's the answer. It's 95.45% of

the curve is below z equals 1.69, or below
x equals 90. So as I mentioned before, but

I'll just mention again, you're supposed to
treat all probabilities to the left of z equals

negative 3.49 as P equals zero. So I showed
you what negative 3.49 looks like in the Z

table. It's like point O two. Well, there's
not much smaller than that. So just, if you

actually calculate z and you get like negative
four, just say the P is zero, okay. Then the

second thing is treat all areas and probabilities
to the right of z equals 3.49, SP equals one

or 100%. So as you can imagine, you know,
3.49, that's at the top of the curve. So if

you calculate a Z and you got like a five,
you can just assume that's 100%, right or

one. Okay, um, so we've gone through how to
calculate z. And we've talked about looking

at probabilities in the Z table. And we've
even talked about manipulating those probabilities

to get certain probabilities. But we haven't
talked about calculating x when z is given.

So sometimes you're actually given a z. And
you are have to calculate the x back

from the Z. In fact, sometimes it's even harder.
Sometimes you're given a probability. And

the probability is not as easy. But you can
use the probability, remember that those little

percents in the middle of the table, you can
go find it in the middle of the table and

look up the Z that keys to it, and then put
it into this equation. And so I'm going to

just give you examples of some real life questions
that you might see, like on a homework or

on a task, probably not in real real life.
That where you need to calculate x, and you

need to use that formula in the red circle.
So let's say I was just bored. And I was wondering,

what is the score the test score on the story
distribution? That is add z equals 1.5? Okay,

so see where z equals 1.5? We never asked
that question before. So let's say I just

out of curiosity wanted to know, what would
the test score be of a student who was at

z equals 1.5. So what I would do is I would
take 1.5 times 14.5, because that's what the

formula says. It's z times the standard deviation.
And then I do that first because order of

operation. And then after doing that, I'd
add the mu, which is 65.5. And I get 87.3.

So the x, the student who got 87.3, that student
got a score, that's add z equals 1.5. Now,

as you probably imagine, people don't go around
asking so much about well, I wonder what that

person's score is at z equals negative 2.3?
Or whatever. They don't usually phrase it

like that. Usually, you see more like a question
like this, which is what is the score that

marks the top 7% of scores? And that's a secret
way of saying, We are looking for the Z at

p equals point. Oh, seven. Oh, so it's like
we turn that 7% backwards into probability.

And we say, we're actually looking for the
Z at p equals point. Oh, seven. Oh, so how

do you do that? Well, I'm going to show you.

Okay,

so we're on the hunt for probability. Point.
0700. Okay, so let's start at the top of the

table here. You'll see we're digging around
in the middle of the table, right? And you'll

see like point oh, that's nowhere near the
ballpark, because we're looking for point

O seven. Oh, so let's scroll up here. or scroll
down, actually. So now we're more we're in

the point O four neighborhood. Here's point
O six. Okay, we're getting close. Well, here

we have a point.

Oh, 708. And that's point oh, eight more than
we want it to be.

Well, here next door, we have point Oh, 694.
And that's only point oh, six less than we

want it to be right, because if it had point
O six more, it would be point O seven. Oh,

so this

is technically closer than this one, because
this is point O, O eight off. And this is

only off by point O six. So we're gonna choose
point o 694. As the probably the probability

of record for this for the top 7%. Only, we're
not going to just choose this, we're going

to figure out what is z at that score. So
what are we gonna do, we're gonna map back

here, negative 1.4. And then we got to go
all the way up, which we can guess is eight.

So it's negative 1.48. So hold that thought.
Okay, we started out looking for the Z p equals

0.0700. And but the closest we got was 0.0694,
and then map to z equals negative 1.48. Now,

what I want you to notice is negative 1.48
is actually on the left side of me. Okay,

so that is the z score at the bottom 7% of
the scores. So we're going to use the positive

version of that see, since we want the top
7%, so we're going to use 1.48. So the opposite

See, and now we're going to plug it into the
equation. So 1.48 times 14.5, which is the

standard deviation plus 65.5 equals 87. So
now at seven is the score that marks the top

7% of the scores. I'm going to do another
exercise for you. That does the this time

the bottom 3% of the scores because this is
often kind of challenging for students. So

I'll just give you a second demonstration.
So as you can imagine, we're going on the

hunt now for z at p equals 0.0300. So let's
go over to the Z table. All right, now we're

getting a little good at this, right? So we're
digging around in the middle, and we're looking

for 0.0300. Okay, and starting at the top,
we're in the 00. department. Oh, here's point

01. Something 02. Okay, we're getting close
to the point 0300. So a point, point 0301.

Could you ask for anything closer? Totally
Perfect. Okay, so that's what we're going

to use for our z is the the Z at 0.0301. So
let's look up that C so that c is negative

1.8. And then we look up eight, so it's negative
1.88.

Hold that thought.

All right. Well, we were on the hunt for P
equals Oh, point oh, three. Oh, and we didn't

find that. But we did find p equals point.
Oh, 301 and the table, and that mapped back

to z equals negative 1.88. Right. And now
we go back to the question, we see that we

want the bottom 3%, so we keep the negative.
Now if I'd asked about the top 3%, we'd lose

the negative we use 1.88 in the equation,
but since we want the bottom 3%, we're going

to keep the negative. Okay, so now let's do
the equation. So x equals and then in the

parentheses negative 1.88 times 14.5, which
is our standard deviation, then plus our mu,

which is 65.5. And the score we get is 38.2.
So 38.2 is the score that marks the bottom

3% of scores, and just be happy your score
is not in there. Okay, now, here's another

challenging hard question. What is the question
on the tester, probably not in real life,

but on a test says what scores mark the middle
20% of the data. And so I put little arrows

on there just to point out well, when they
say middle, they mean, it's hugging

the meal,

it's actually assuming that there's gonna
be 10% on the right side of the meal, and

10% on the left side of the meal. And so how
you start to do this is you figure out the

z score for one minus point two, which is
the 20% divided by two, which equals four,

right? So then after that, you know, because
one minus point two is point eight, and point

eight divided by two is point four. So we
get this point four. So we go find the z score

at point four, which you're good at using
the Z table now. So uh, so I'm, you know,

looked around, and I found point 4013, in
that, digging around in the middle of the

Z table, and that map back to negative z equals
negative point two, five, right. And so that

is then what I would put on for the lower
limit on that one, and then z equals point

two, five, the positive version goes on the
other side. So once you figured out both of

the Z's, the Z on the left and the Z on the
right, you just have to put them through the

equation. So for the left side, we use the
negative z. And for the right side, we use

the positive Z. And that's how we get our
limits. So what's for is mark the middle 20%

of the data 61.9 and 69.1. It's not weird
how that worked out. But anyway, 61.9 and

69.1. Mark the middle 20% of the data. I didn't
totally didn't do that on purpose. It just

worked out that way. All right, I can't believe
you made it through all this. I'll bet your

brain is ready to explode. So now is a good
time to talk about just a little review. Just

help me come down a little bit from this whole
really intense lecture. Okay. So first, I'm

going to do a little Z score quiz game show
style stuff here, right? So if you ever get

the question when you're on the test, and
you're like, Oh, my gosh, where is x? Where's

x? Well, if you can't find x, it's usually
in the question. So usually, the way these

questions go is somebody like maybe me, we'll
put a mu and a standard deviation at the top

of the question. And then there'll be like,
maybe five questions about that pertain to

that mu and that standard deviation, but they
asked about different axes. And when I would

teach this class, a person, you know, people
will come running up to me in the middle of

a test, which you probably shouldn't do. And
they would say, where's the x? Where's the

x you gave me you know? These pieces of the
equation but I can't find the x. And I'd be

like, walk on the question. Look in the question,
you know, because I don't want to give it

away, and then they'd all run back to their
seats and find it. So that's so if you're

wondering, your panic and where's x? Look
in the question, it's usually in the question.

Okay, so let's say you find an X, and what
do you do with an x? Okay, and you're stuck

with an X, what do you Well, usually, what
you have to do is calculate a z score. So

remember, if you've got an X, you probably
have a mu and a standard deviation, you can

calculate a z score on that. So if you're
panicking on a test, and you have an x, I

mean, Sandy nation, just for fun, calculate
a z score and see if it gets you anywhere.

Okay, well, let's say you have a z score,
what do you do with a Z score? Well, you always

look it up, right? I mean, if you're, if you're
going this direction, if you're getting if

you started with an X, and you get a Z, you
got to go to the Z table with. Okay, so that's

your next step. So if you're doing all this
work, calculate a z score. And then you're

done. You're like, Oh, my gosh, what's my
next step? Go look at the Z table. Well, what

is the question asks for an x, right? Well,
remember, we have a whole formula for that.

So use the x formula. So if there's no x anywhere,
and it's asking for an x, then use the other

formula, use the

x formula?

And what if the question gives you a P, or
I just said p for probability, but it could

be a percentage, like Remember, the top is
7%, and the bottom 3%? Well, if they give

you a percent, just start digging around in
the middle of the Z table, just start digging

around looking for that person. Because once
you start digging around, you realize that

map's back to a z. And then you can get into
the groove of using the x formula, and you'll

probably get yourself out of this pack. So
here are some final tips and tricks for getting

z scores and probabilities, right? And I've
said this one before, draw a picture. And

what do I mean by that graph out the question,
draw the curve, draw the line from you, which

goes in the middle. And where the X goes above
or below the mu, just start with that it doesn't

have to be the scale. But mainly, you want
to get those elements in there. There's 1x

shade, the part of the curve wanted either
above the X or below the x, you know, just

color it in. So that you get an idea of Do
you want the big part, the one that's greater

than 50%, or the little part, the one that's
less than 50%? If there are two x's, then

shade in the area wanted, which is usually
in between them. If it's a calculate the x

question, put where the Z or the P is. So
if it was like the top 7%, you could shade

in the top little part of the curve. If it
was the bottom 3%, you could cheat in the

bottom little part of the curve. So make this
picture and do it at the beginning. Okay,

then, note that x is usually in the question.
If you can't find x, and you're trying to

do the Z formula, and you're saying, Okay,
I'm trying to make a z score. That's what

it asks for. I'm trying to find a probability.
That's what it asks for looking the question,

and you'll probably find the accent there.
A big problem that I see is people mistake

little Z's for peace. Now, obviously, if you've
got a Z, that's like negative, you know, a,

p can't be negative, a probability can't be
negative. So you won't make that mistake.

Even if it's like negative point two, five,
right? You won't make that mistake. And if

the Z is bigger than one, you won't make that
mistake. So if you see a z equals 2.5, you're

like, obviously, that's not a probability.
But when you have a little BBC score, that's

between zero and one, like point O two, three,
it looks a lot like a P, but it's still a

z. So a lot of times people get a little lazy,
like they hate using the Z table, and

then they calculate the z score, and it's
really little, so they don't look it up. Don't

be fooled. You still have to look it up. So

if you're calculating z, you need a little
baby z like that it still is he still go look

it up. Okay. Then finally, remember how step
one was draw a picture. And I went on and

on about that. Step 99. Or the last step before
you're done with the question is check your

logic against that picture. So if you shaded
a big part of your picture, your probability

should be bigger than point five, or 50%.
If you shaded a little tiny part of your picture,

and you're getting like point nine, five,
something, you know that that's wrong. So

please check your logic against the picture.
Before you say that you're done with your

question. Okay. So you made it through this
long lecture about z, and about probabilities.

So I gave you an introduction to the standard
normal curve into those two Z score formulas.

I showed you how to calculate z scores, and
how to look at probabilities. And I also showed

you at the end, how to calculate x if given
a z score or a probability. Okay, and all

I want to say is, unfortunately, those students
those pretend students on that distribution,

they were none of them got 100% Okay? That's
not the case in our class, a lot of times

people get 100% on the quizzes. That's why
I can't use your grades as examples. Okay,

so good luck on the quiz. Well, hello, it's
time for statistics. It's Monica wahi, your

library college lecturer back with chapter
7.4 and 7.5 sampling distributions and the

central limit theorem. So at the end of this
lecture, you should be able to state the new

statistical notation for parameters and statistics,
for two measures of variation.

Name one type

of inference and describe it. explain the
difference between a frequency distribution

and a sampling distribution, describe the
central limit theorem in either words or formulas,

and also describe how to calculate the standard
error. So, here's your introduction to this

lecture. And as you can see, I must 7.4 and
7.5. Together Again, they felt like a natural

fit. First, we're going to review and maybe
overview on parameters, statistics, and also

inferences, we're going to just talk about
those ideas, because that will sort of easy

into the next part, which is where we start
talking about sampling distribution, which

is the new concept here. Okay. And then we'll
go on to talk about the central limit theorem.

And finally, I'll do a little demonstration
of how to find probabilities regarding x

bar.

So if you're not really sure about what that
means, don't worry, you should be able to

understand it at the end of this lecture.
All right, here's the first part, parameters,

statistics and inferences. And this is the
review and overview I promised you. So if

you remember from a long time ago, a statistic
is a numerical measure describing a sample.

And a parameter is a numerical measure describing
a population remember s s sample statistic

p p, population parameter, you probably remember
that. Okay, so we have different ways of notating

these. So if you look under measure, like
you see me right, and if it's a statistic,

it's x bar, and I say x bar on this on the
slide sometimes because it's hard to make

that little line always be positioned above
the x. So I'm just lazy to say x bar. And

then under parameter, it's that that new symbol,
so it's pronounced a meal, but it looks like

that thing on the slide. All right, um, the
next two variants and standard deviation,

remember how they're friends. And so the statistic
version is the s for variance, it's the s

with the little two up there, the exponent,
because you know, it's standard deviation

to the second is variance in the square root
of variance is a standard deviation.

So that's why they have s and then S to the
second for the statistic, okay. For the parameter,

it's that lowercase sigma symbol. And that's
it's that to the second when it's variance,

and it's just without the exponent, when it's
just the regular parameter of standard deviation,

right.

And you're used to seeing these on the slides.
This is just review. I'm also in mentioned

in the book proportion is p hat, and then
the parameter is P. But I don't really go

into that. I just wanted to do a little shout
out to it. Okay, let's think about the word

inference, like infer, like, if somebody implies
something, maybe you'll infer it. Like, he

implied, it would be hard if I came over late
that night. So I inferred that I shouldn't

come over late then. So like here, you know,
you may have heard the term where there's

smoke, there's fire. And so you see this on
the slide, there's a lot of smoke. Is there

fire, though, is that smoke coming from fire?
Because if you look at it, it probably could

be coming from fire. But there's sort of this
outside chance. It's not what we think it

is, like maybe, you know, I have if you've
ever used a fire extinguisher, they make all

this phone come out. Maybe it's that, you
know, or maybe it's like, if you've ever had

dry eyes, and then that makes a bunch of smoke.
Maybe it's not fire, right? So where there's

smoke, there's fire. That's an inference.
Well, let's see

if it's actually fire,

right. But we weren't sure we thought it was
likely to be fire. But we weren't sure. And

so there's inference is something that you
do in statistics, because you use probability

to make these inferences because you can't
see the fire. You can just see the smoke and

you're not sure, right? So there's three different
kinds. I'm going to talk about the first kind

of estimation, where we estimate the value
of a parameter using a sample. So the sample

is kind of like the smoke and the parameters
the fire we can't see. So we estimate

Okay, and we're going to talk about that in
chapter eight more. A second time, type of

inference we do is testing, where we do a
test to help us make a decision about a population

parameter. In other words, we don't know one,
but we want to make a decision about it. So

we do a statistical test. And we're not going
to get into that, that's in chapter nine.

Finally, there's regression, where we make
predictions or forecasts about a statistic,

that's a third kind of inference. And we actually
already did this in chapter 4.2. So the reason

why I bring up all of this is that estimation,
which is going to be in chapter eight, and

testing, which is going to be in chapter nine,
but we're not going over chapter nine in this

class. But um, but if we were, you know, you'd
have to know this because in this lecture,

I'm going to talk about sampling distributions
in the central limit theorem. And you need

to grasp those things in order to do those,
these two things on the slide that with the

box around them, estimation, and testing.
And so that's why I'm bringing this up now.

Okay, so now we're going to move on to talking
about sampling distribution, and how it's

different from a frequency distribution. Alright,
so let's just remind ourselves what a frequency

distribution actually is. Okay? So remember
that from a long time ago, what you would

have is a quantitative variable, you'd make
a frequency table. And then you use that to

graph the histogram, right. And here, I made
an example down there of frequency histogram

that shows a normal distribution. And so that's
what you would do, you know, step two would

be draw it. And then you see the shape and
figure out what the distribution was of that

quantitative variable, or that x, okay, because
each one of these is an X, like the middle

one, it's almost 30 X's that are in that frequency.
Okay, now we're going to talk about sampling

distribution, it's a little more complicated.
In a sampling distribution, you start out

with a population, that's the first thing
is you're dealing with population, then you

pick an N, of a certain size, like you pick
a number, that you're going to have your sample

size B. And then you take as many samples
of that size as possible from the population.

And then you make an x bar from each of the
samples. So there's a ton of samples, right?

Because and I'll show you a little demonstration.
So you can really wrap your mind around how

many different samples that can be. But each
one is going to have an x bar. And then you

make a histogram of all those x bars. So like
I said, I'm going to just kind of show you

what I'm talking about. So we're going to
imagine this is a population of people. And

we're going to imagine we're going to talk
about BMI or body mass index, just so you

can wrap your mind around this. So you start
with this population, let's decide on an N.

How about five five is good, right? So now
what the deal is, is I'm trying to take as

many samples of n as possible from all of
these people on the slide. So here's our first

sample we took, and we got an x bar for BMI
of 23. From these five people. Well, let's

try these five people. Now, look, we double
dipped with that first one, okay, but we get

this x bar of 21. And we can keep going. And
actually, there's gonna be a ton of these,

right, there's a ton of different ones. But
it's finite. I mean, at the end of the day,

there's only so many groups of five, I can
get out of this population on the slide, and

each group of five is going to have its own
x bar. So I could write down every single

one of those x bars I get for every single
group of five I can make out of this. And

then I can make a histogram of all the x bars.
And, of course, I'd start with a frequency

table. But look at the frequencies, they're
huge. That's because you can get just a ton

of samples out of one population. And so what
you'll see is if you make a histogram out

of that, it looks normally distributed, it's
just that the frequencies are really high,

because there's a whole bunch of different
samples you can take. And remember, this is

a frequency histogram of x bars. This is each
one of these frequencies is an x bar that

you got out of a group of five you could take.
And so that's what the sampling distribution

is, it ends up looking like a histogram, but
it's a histogram of all the possible x bars

you could get from all the possible samples
of whatever end size you picked from the population

that you

have.

So uh, so this is the fancy way, the official
statistical way of saying it is a sampling

distribution is a probability distribution
of A sample statistic, in this case x bar

based on all possible simple random samples
of the same size from the same population.

So that's what makes it the sampling distribution
and not a frequency distribution. And so in

the next section, so you're probably like,
Okay, great, that's wonderful. You just explained

that. But in the next section, we're going
to talk about the central limit theorem, here

comes a theorem, right. And there's a proof
for the theorem. And you need to understand

this concept of sampling distribution for
inference in order to understand this proof,

so I just had to go through this. Okay, now
we're on to the central limit theorem, and

how it's used for statistical inference. So
I'm gonna start by explaining it in words

and see that sampling distributions over there.
So this is the words around the central limit

theorem, it says, For any normal distribution,
and remember, we're talking about a normal

distribution here, the sampling distribution,
meaning the distributions of the x bars from

all possible samples, like we just talked
about, is a normal distribution, meaning it's

not skewed, it's not my model, whatever, it
looks kinda like what is on the slide. Okay.

And then to this is important, the mean of
the x bars is actually mu. So I had a student

who would say, Oh, the x bar of the x bars,
is mu. And that's actually true. If you actually

did the thing I described, which don't try
it at home, because you'll be up all night

taking samples, okay. But if you did, if you
actually got all samples of five from a population,

and got all their x bars, and you made a mean
of all those x bars, you'd get mu and how

you could check it is, of course, just easily
taking a mean of the entire population like

that would have been the easy way to do it.
But no, if you do it this way, where you get

every possible x bar for a particular sample
size, and then you make an x bar, those x

bars, you'll get meal. So that's, you know,
it's a proof. So that sounds like a thing,

that would be inappropriate, right? Now, here's
the next part three, the standard deviation

of all those x Mars is actually the population
standard deviation divided by the square root

of whatever and you picked. So in other words,
if you have the whole population data, and

you just found out the standard deviation,
you just have the standard deviation. But

if you did this thing with the x bar, where
you took all those x bars, and you found the

standard deviation of those x bars, that would
equal the population standard deviation divided

by the square root of whatever n, you use
to get all those x bars, again, sounds really

poufy In theory, but that's the third part
of the central limit theorem

in words.

And so here's some people like to look at
it from a formula standpoint. So you'll see

on the right side of the slide, in this little,
these little formulas, that N means the sample

size. And remember, I picked five, you could
pick a different one, right? And mu is the

mean of the x distribution, meaning the population
mean, right. And then that population standard

deviation symbol is the standard deviation
of the x distribution mean the population

standard deviation. So we look on the left.
Now this is just a formula version of what

I just the mu of all the x bars that you could
get from a particular sample in a particular

population is going to equal the mean or the
population. And the standard deviation of

all those x bars is going to equal the population
standard deviation divided by the square root

of whatever n you picked. So now, I just want
to point out the Z thing. We've been doing

this z thing, right, but we've been doing
it with 1x. Now, if you imagine grabbing a

bunch of x's, in other words, a sample, this
is the formula you're going to be using, which

is x bar minus mu over the standard deviation
divided by the square root of n, right? And

so that's kind of what we're moving into here
is what happens if you get a sample and you're

looking at x bar, not if you just grab 1x.
And you're looking at that. So I wanted to

point out, first of all, that this whole thing
is only supposed to happen if your n is greater

than 30. Okay? Otherwise, you shouldn't really
be doing this. Then the second thing I wanted

to point out is that this piece underneath
and the lower part of the equation, that's

called the standard error, they named that
piece. And part of the reason why I like that

they named that piece separately, is I usually
make that piece before I even do the equation.

So I just have that number sitting around
because, you know, there's a square root underneath

this standard deviation, and that whole thing
is underneath another thing so it's hard to

do all that dividing. So I usually just make
that standard error first, by taking the standard

population standard deviation divided by the
square root of n and just have that number

and then later I use it in this z equation.
So that's two things I wanted you to notice.

So I brought that out on the slide. Okay,
here's more on the central limit theorem.

So if the distribution of X is normal, then
the distribution of x bar is also normal.

So we look at the top, that's an example of
just an X distribution. And then if you go

do that thing, we take all those samples,
and you get all those x bars. And then you

make the histogram, you'll see the pink one
down, lower. Next bar distribution,

this is just a pictorial example.

But even if the distribution of X is not normal,
as long as there's more than 30, and is more

than 30, the central limit theorem says that
the x bar distribution is approximately normal.

So remember, a lot of that hospital data we've
been looking at, like a hospital beds in a

state, often you'll see a skewed distribution.
But if you have more than 30, hospitals, then

it what you could do is you could pick n n,
and take n bigger than 30. And take a bunch

of samples and get a bunch of x bar, it's
not just a bunch get all of them all of the

possible ones. And then when you if you made
that x bar distribution, even though the hospital

beds would be skewed, just as an X distribution,
their x bar distribution would be normal.

And that's one other important piece of the
central limit theorem. That's one important

piece of that proof is that all of those x
bars that you get, will end up on a normal

distribution, even if your underlying distribution
is not normal. So long as the end you're picking

is greater than 30. And finally, that leads
to you know, proofs are they build on each

other, that leads us to the concept that a
sample statistic is considered unbiased, just

unbiased, right? It's not perfect, but it's
unbiased. If the mean of its sampling distribution,

equals the parameter being estimated, in other
words, the fact that the x bar of the x bar

is is mu, means that an x bar is going to
be unbiased. It might not be mu, it might

not be exactly the same as the population
mean. But it will be unbiased. It's not a

biased representative of mu. All right, now
let's move on to finding probabilities regarding

x bar. So for those of you who want to actually
do something and apply something and stop

thinking about theory, let's go. Okay, but
let's remind ourselves, what are we doing?

Right? What are we doing? Well, what were
we doing in chapters 7.1 through 7.3, we were

looking at having a normally distributed x.
So we have this population of quantitative

values that were normally distributed. And
we had a population mean a mu, and we the

population standard deviation. And we kept
doing these exercises, where we were finding

the probability of selecting a value from
that population and x from that population

above or below a certain value of x, right.
And so we were looking at the probabilities,

and we'd look up the z score in the Z table
probabilities. And so basically, what we would

be doing is converting m x to z, right. And
we use this formula here to convert x to z.

So whenever we add an x, we could put it on
the Z distribution, and we could figure out

the probability. So here's what's different.
Now, you'll notice the first thing has not

changed, we're still talking about normally
distributed x's, we're still talking about

a population where we have a mu and a population
standard deviation. But now we're not just

grabbing 1x. From that population, we're grabbing
a sample. And because we're grabbing a sample,

we have to pick an N. So the N is going to
be different each time, right? So we're grabbing

a sample of the population. Well, how do we
boil that down to one number? Well, we're

taking the x bar are the mean value from that
sample. And that's what we're doing. The Z

score is that x bar instead of the x, because
we're taking a sample, so when you see the

formula below, you'll notice that the other
one just had x in it, because we only had

one, this one has x bar, and because we have
a sample, you also notice that downstairs,

what we had before was the population standard
deviation, but now

we have the standard error. Remember I talked
about that the population standard deviation

divided by the square root of n, that's where
n comes in, because it's going to matter which

what and you have to make the Z come out right?
Alright, so now that we're reminded of what

we're doing, we'll just explain how to do
it right. So let's say you do have an N, right,

and you have an x bar, like you grabbed your
n and you got an x bar, you can convert that

x bar to a z score using this formula, where,
of course, you have to be told the population

mean and the population standard deviation,
but then you'll have your x bar and you'll

have your n. So you can do the whole equation.
And then you'll get to see and guess what

you do. What do you do with a Z, you look
it up. So you look at the probability for

the z score in the Z table. Like in chapter
7.2, and 7.3. Only, this is just about x bar,

basically. So um, and then I thought, what
I would do is walk you through two examples.

You're already kind of good at this, because
this is not too different from 7.2, and 7.3.

But I just want to walk you through it, because
it is a little different when you have a sample

versus just 1x. Okay, so remember our poor
chemistry class that I was in when I got to

73? Well, remember, we were assuming it was
100 Student class. So there were 100 students

in the class and equals 100 in the class capital,
right, because they're the population. And

then if you look on the slide, you'll see
the mu of their scores was pretty bad. It

was 65.5 on 100 point test, and the population
standard deviation was 14.5. So this was the

population of this 100 Student class. So I'm
going to do some exercises here, let's say

we're going to pick a, we have to pick an
N bigger than 30. So we're going to pick an

N of 49. Right? Now, I'm coming up with a
little scenario here. To pass the class students

have to get at least 70, which is a C. So
let's pretend this is the question, what is

the probability of me selecting a sample of
49 students with an x bar greater than 70?

Notice how we ask the question a little bit
differently. What's the probability of me

getting a set of 49 students such that their
x bar is greater than 70? Does not kind of

remind you of the central limit theorem, where
we had to go back and get a like an N a five,

we got different ends of five? What what's
the probability of me getting one of those

samples that has an x bar in the greater than
70? That's the question, right. And I drew

this out here, remember our old z distribution
with our also our x distribution, and I kind

of drew where somebody is. But I wanted you
to point I wanted to point out for you, the

probability for an x bar is going to be smaller
than for x, because you're going to have to

do a lot of work to get that x bar to be above
70. Right? So here we go. So I'm just going

to remind you that the equation at the top
and the equation at the bottom are the same

equation. I'm just using the term assay for
the standard error. And I like to calculate

that separately, like I told you, so I like
to do that first. So we're going to do that.

And how do we do that? Well, the end was 49,
right? And I'm the population standard deviation

is 14.5. So that's where we get this, this
number, the standard error of 2.1. So now,

let's calculate the Z. All right, here's z.
So z is our x, which is our x bar, which is

70 minus 65.5, which is our mu, divided by
our prep cooked standard error, which is 2.1.

And we get a Z of 2.17. So we're tempted to
look that up. But let's look at our picture.

So here's our z distribution. And what we're
going for is this little piece at the top

right above 2.17. So that's a little piece.
So we got to look for that right? Let's go

look. So because we're going to go for the
piece at the top, we're going to use the opposite

z. There's remember two ways of doing this.
But everybody seems to prefer the way where

you use the opposite z if you're looking for
something to the right. So we're going to

use negative 2.17 to get a little piece, right?
Because when you look that up, I'm not going

to demonstrate you guys are good at this now.
You get P equals 0.0150. If you were to look

up 2.17, then you'd get the big piece. So
that's why we do this. And so then the answer

is, remember the question was what is the
probability of me selecting a sample or a

set of 49 students with an x bar that's greater
than 70. And remember how this real test really

sucked. I mean,

people that mu was 65.5. So it was pretty
hard to get a high score. So the probability

was pretty low as point 0.0150. Or if you
do that Present version 1.5%. Okay, now we're

going to try a different one. That one was
asking what is the probability of me selecting

a sample with an x bar greater than a certain
number? Now we're going to talk about the

probability of selecting a sample with the
x bar between two numbers, right? So again,

we're back with our poor student class that
with this terrible chemistry test, this time

I decided to choose the end of 36, you'll
notice that I always choose perfect squares

for ends because you have to take the square
root, and I'm just lazy. So okay, here's our

question, what is the probability of me selecting
a sample of 36 students with an x bar between

60 and 65. And just I drew this picture up
here to remind you that, that's gonna be on

the left side of meal, you know, we're going
to be dealing with negative Z's right. And

so we have to remember when we would have
two axes, back in 7.2, and 7.3. Well, this

is now a situation where we have 2x bars,
so you just got to name them x bar one and

x bar two. And, again, I show you this demonstration,
you know, these red arrows, but the probability

for x bar will be smaller than for x, because
it's harder to get a whole group of people

together to give you an x bar in between a
certain place. Alright, so this is not new,

these are the same formulas I showed you before,
I just want to emphasize that making your

standard error first, can really help you
as you move along through these problems,

it just makes it a little easier to calculate,
especially in this case, where we're going

to use the standard error twice. So again,
what we do is we take, this would look exactly

like the last standard error, but it's different
because our n is different. So this time,

our standard error comes out as 2.4. And what
I just want to remind you is that the more

and you get, the bigger that square root of
n gets, I mean, n gets bigger, the square

root of n gets bigger. And that's then the
smaller the standard error gets. So you can

make the standard error really small, if you
just get a lot of n, right. So here's z one

and z two, I put them both up there. But we
can just walk through this, you know, x bar

one is 60. And x bar two is 65. Because it's
between 60 and 65. So you see that, um, you

see what's going on in the slide. And like
I told you, you know, these were both of these

x bars are below the mu. So they're both kind
of negative Z's. And so we've got our negative

Z's. And that now we have to just remind ourselves,
well, what are we doing, right? And so you

see, z one is at negative 2.28. So that's
a little piece at the bottom, we're going

to want to trim off. And then the big piece
at the top for z two, that starts at negative

point two, one. So that's just remember, the
picture is really helpful. So now we're going

to go deal with the probabilities, right?
So for z one, we're looking at something to

the left, so we just leave the Z alone and
go look it up. And that's p equals 0.0113.

For z two, we got to flip the sign because
we have to use the opposite z, because we're

going for the right, so that was the probability
two and we can check that see, because we

can see that's more than 50% of that shape.
So it's point 5832. Okay, so we got our probabilities

now. And like just like last time, we got
to take one minus both of those pieces, right?

And then we get the probability in the middle.
And that's the probability of drawing us sample

of 36 students with an x bar between 60 and
65. And I just to translate that to the answer,

the probability is point 4055. Or if you rounded
it, you know, when you like, percents, you

could say 41%. So in conclusion, we reviewed
the parameters, and the statistics, and those

notations. And we talked about inferences
and what we're doing with inference. Next,

we talked about what a sampling distribution
is, and how that's different from

a frequency distribution. So you can tell
you know what's going on with that. Then I

presented to you the central limit theorem,
which may have been kind of confusing, because

you know, theorems always are, they're always
about different principles and about different

things equaling each other. But because of
the central limit theorem, we then have permission

to do the operations we're doing after that,
which is finding probabilities regarding x

bar. The central limit theorem says that,
you know, this is how the world works. So

you get to use the standard error, and you
get to do these kinds of calculations. So

now, you know how to in addition to finding
probabilities regarding x, you can find probabilities,

we got x bar. Don't you feel smart