All language subtitles for 968.English

We humans,

have a keen eye for

visual treat

and love a good eye candy.

That being said, statisticians

were having a hard time

with getting people to listen

to their very important

and relevent data.

Frankly speaking,

even though tables

are easier to consume,

it still does not taste good.

Does it?

So what

to do?

A lot of data

that we deal with

in the real life is comparative.

As in comparing 2 things.

It can be about

which student is tallest

or which candy is cheapest

and so on.

This is where

graphical representation

comes to our rescue.

Graphical representation of data

allows us to understand

the data much more easily

and intuitively than a table.

Our aim here,

is to throw some light

on 3 major types

of graphical representation of data.

That is the bar graph,

the histogram and

the frequency polygon.

You already know

a few things about graphs

from your earlier classes.

Let's build on that.

Let's represent table of heights

in the form

of a bar graph.

Let's bring up the table

of ungrouped data here.

To draw the chart,

I'll start off by drawing

a flat horizontal line

called the X axis.

Where you represent

different height values

of all the students.

Now, as I

go through this data,

I start to add a dot

above the X axis.

So first, I put a dot

at 195.

The next is at 175.

The next at 170

and I keep continuing.

The fifth student

is at 185.

And so is the sixth student.

So I add a dot above that.

This can continue

till I exhaust the complete data.

So if you observe,

the Y axis represents

the number of students

or the frequency.

Because this is called

a bar graph and not

a dot graph,

instead of you using dots

like you just did,

you can start to draw

rectangular bars and extend it

till your corresponding data point.

For example, there is just

one student with the height of

130 cms. So I extend the bar

till it reaches the level of

1 on the Y axis.

The next data value is

135. Has a frequency of 2.

So I extend the bar

till it reaches a value of

2 on your Y axis.

Moving on, we have 155

which appears 7 times.

So the bar for this value

extends till it reaches 7

on the Y axis.

162 goes upto 6.

168 goes up to 4.

Finally 195,

with the frequency of 1.

Remember that the thickness

of all these bars

that you see

is actually of your choice.

But for the sake of clarity,

you tend to maintain

all of them

as the same thickness.

From this you can easily tell

the number of students

that have the same height

by looking at the

height of each of these bars.

This becomes all the more necessary,

when we ask questions regarding

a particular height

and also how many students

have the same height.

We apply the same logic

to grouped data as well.

In this, we have

heights of 60 students

grouped into classes of 10 each.

So we drop the axis again.

This time we have

the classes on the X axis

and corresponding frequencies

on the Y axis.

The class of 130-140

has a frequency of 9.

So the bar extends from

X axis, to reach

a level of 9

on the Y axis.

Similarly, for the rest

of the frequencies.

Making data visual,

makes it leads better

to understand it. Doesn't it?

A similar representation can happen

using a histogram as well.

Let's dive in.

A histogram is just like

this bar graph.

But I will have to make

a few changes here.

Just like a bar graph

we represent the height

on the horizontal axis

but using a suitable scale.

Scale here, becomes very important

because the area

that this bar covers

in a histogram

is very very important.

We can choose the scale

as 1cm equivalent to 10cms.

So each class occupies

a width of 1cm

on this graph.

Also since the first class interval

is not starting from zero

but a fixed non-zero value

we show it on a graph

by marking a Kink

like you see here.

As this has a break

on the axis. Next.

Unlike the bar graph

there are no gaps

in between the rectangles

of the graph.

So, I will have to

knock off all of these gaps

and will have to keep

only the lower class limits

on the graph.

Technically, it is one solid figure.

What you see now

is called a Histogram.

One important thing

that you will need to

keep in mind about a histogram,

is the area of the graph

plays a very crucial role.

In fact, the area of the bar

is directly proportional

to the frequency of that data.

Also the sum of areas

of all the bars

is equal to the

total frequency of all the classes

in the table.

Till now we have dealt with

classes of equal sizes.

What if I have

different class sizes

on the same histogram?

That is, what if

I have to put all students

with heights less than 150

in one bracket

and anyone with heights

more than 170 in another bracket.

Absolutely arbitrary.

So that means

class interval 130-140,

140-150 get clubbed

into one class interval

of 130-150 along with their

respective frequencies.

Likewise class intervals of 170-180,

180-190, 190-200

all get clubbed

in a class interval of 170-200.

And in between, we have 150-160

and 160-170.

That remain as is.

So, that means, now

you have a new table

with classes of different widths.

Tthe first class width is 20.

That is 150 minus 130.

Followed by 2 class widths

of 10 each.

That's 160 minus 150.

And the last one

with a width of 30,

which is 200 minus 170

that you see.

If I were to draw

a bar graph here,

this is how it would look.

For a histogram

on the other hand,

I mentioned that the

areas of the bars are crucial

for accurate representation.

We need to pay attention

to the width and height

of these bars here.

So, the width of

all the 3 classes are

different. Remember

I told you that the

area of a histogram

has to be proportional

to the frequency.

So how do we do this?

We need to bring

all the frequencies in line

with the minimum class width.

The minimum class width here is

10.

The length of the rectangles

are to be modified

to proportionate this class size.

For instance,

when the class size is 20,

as is the first case.

The length of the rectangle

will be 16 times 10

divided by 20,

which is going to be

equivalent to 8.

This is simple cross multiplication.

This way the total frequency

will be 16 in this range.

The next 2 groups

the class widths are the same

as the minimum class width.

Hence you don't need to

change anything.

The last one however,

goes through the same treatment

as the first one.

In this instance,

the class size is 30

and the frequency is 8.

So when the class size

becomes 10,

the length of this rectangle

will be 8 times 10

divided by 30.

That is 2.666

This histogram can now be said,

to be proportional

to the students

per 10 cm interval.

Even though a bar graph

and a histogram look alike,

you might have noticed already

that there are a few differences.

In fact if I bring them together

unless you are a statistician,

chances are,

that you will get confused.

This exercise that

we will do now,

will help you sort out

this confusion.

If I ask you to

collect data about language preferences

of the students and

add it to our

original table.

Now I will be able to

draw a bar graph

out of it.

Now let's try to make

a histogram out of the

language data

that we have collected.

Is that even possible?

Hmm. No it is not.

Infact the data that you collect

can be split into qualitative

and quantitative data.

If you're looking at

colors of the car

on the road,

then the color of the car

which is a data

which is of the qualitative kind

because this describes the

quality of that particular data.

Or if I ask you

the flavor of ice cream

that you like,

that again is a qualitative data.

On the other hand,

data such as heights,

weights, roll numbers, etc.

are data that are

represented by numbers.

Here height is 160 cms tall.

160 is a quantitative data

since it refers to

numerical data.

From the examples

that we have solved before,

you can see

that we can represent both

qualitative and quantitative data

on the bar graph.

Where as we can represent

only quantitative data

on a histogram.

So the next time,

you need to make a graph

be sure to analyze

what kind of data

you are trying to represent.

Now let's start making

a difference table out here.

And let's start populating

the differences as we go about.

Let's bring back the

graph of heights

from the bar graph section.

Now we see that

on the X axis,

each data point is represented

individually. For example,

a student's height of 130 cms

is represented individually

as 130 cms on the X axis.

This kind of data representation

individually is called Discrete data.

And we also

know that we can

construct a bar a graph

using grouped data as well.

Grouped data here, refers to

when a data point is represented

not individually but as a

continuous range of values.

In case of discrete data,

we can have gaps

in between the values

of data points

on the X axis.

The data that you collect

can again be classified

in one more type.

As continuous and discrete data.

When you're talking about

discrete data, there can be

gaps in the data

that you collect.

For example 130 cms

and 135 cms as heights of students

has a gap of

5 in between them.

And when you're talking about

continuous data,

there cannot be these gaps

that you see here.

So, when it comes to a

bar graph, you can represent

both continous and discrete data.

But in a histogram

you can represent only

continuous data.

This is another reason why

bars of a bar graph

are separated by a gap,

since they are discrete values.

Whereas in a histogram

all the bars are clubbed together.

We also cannot

reorder this data

in case of a histogram

due to continuity of the data.

Let's add these 2 points also

into our comparison chart.

Using continuous data means

that the classes have to be

ordered on the graph

as the appeared to us.

On the other hand

having discrete data

in the bar graph

allows you to arrange the variables

in anyway you want to.

When I'm drawing a bar graph

the order in which

I show the elements

on the X axis

is not a problem at all.

I can first show 130-140.

Then show 150-160.

And then I can have 140-150.

But when it comes to a

histogram, I cannot

reorder the data.

This is obvious because

we are dealing with

continuous variable.

This is one more

for the comparison chart.

As you've already seen,

the spaces in between the bars

are not present in the histogram.

It essentially looks like

one big block.

Also the width of the bars

need not be the same

when it comes to a histogram.

Also remember,

that the area of the bar

plays a huge role

in a histogram and hence,

we need to maintain uniformity

of class width through out.

But in the case

of a bar graph,

the width of the bars

are immaterial

to the interpretation

of the bar graph.

There is yet another visual way

of representing quantitative data

and its frequency.

It's called the frequency polygon.

Let's consider the histogram

that we initially constructed

with equal class intervals.

Let me mark this point,

which is the midpoint

of the class interval

of 130-140.

I will call this point

as the class mark.

So class mark is a

mathematical way of saying

mid-point of class interval

which we obtained

by adding the upper

and lower limits of a class

and dividing it by 2.

If we consider

the class interval of 150-160,

its class mark is

150+160/2

which is going to be 155.

Next I will highlight

the class marks

for all other class intervals

as well.

For a frequency polygon,

all I have to do

is to connect

all of these dots.

Or connect all of these

class marks. Well.

I said frequency polygon.

But what is a polygon?

A polygon is a

multi-sided shape.

But before all

it is a closed shape.

So how do we get that?

We add a class interval

before the first one

in the data

and do the same

in the other end

of the histogram as well.

Since, the first class interval

is 130-140 we add another

with 120-130.

This class interval will ofcourse

have a frequency of zero,

since it is not

represented in the table.

We just have to

mark the class mark

for this group. That is

130+120/2

which is going to be 125.

And then we are done.

We can now connect the line

to the X axis.

Doing the same

on the other end,

we add 200-210

to the frequency polygon graph.

Marking the class mark as 205

and closing the figure

at the both ends

gives us the frequency polygon.

Instead of drawing the entire

bar of a histogram,

you just mark the frequency levels

with the Y axis

at the class mark.

Just like a histogram,

frequency polygon's total area

is directly proportional

to the total frequency

of the table.

For the sake of convenience,

let's bring back the histogram

that we drew

in the previous sections.

Let's take the graph

where the frequency polygon

is drawn over the histogram.

So if I join the class marks

you can see

that the chunks of area

are being leftout of calculation.

There are also a few

empty areas

inside the frequency polygon.

To prove to you

that the area of the

frequency polygon

and that of the histogram

are the same,

I will cut the part

which is outside the line.

Flip it all over and see

that it fits exactly

into the empty area here.

The same can be done

for all the bars.

So eventually, we see that

all the triangles

ejected by the line we drew

are included within this

frequency polygon. And hence,

we can visually say

that the total area

of the frequency polygon

is equal to the

total area of the histogram

made by the same data.

Also the area

is proportional to the frequency.

So till now,

we have learnt about

raw data and how

unless it has context,

it is useless. Raw data

can also be made useful

by processing it.

We process data

by means of statistics.

Using methods such as

creating a frequency distribution table.

Frequency is the

number of times

a particular data

appears in a data set.

When the number of heights

are considered individually

we call it ungrouped data set.

It was too much data

to deal with.

So we then

clubbed the heights to create

a grouped data and a

grouped frequency distribution table.

We then decided

that numbers are all together

too boring,

and came up with

graphical methods of representing data.

This includes bar graphs,

histograms and frequency polygons.

Bar graphs is an excellent

comparative tool

and is used mostly

in non numerical context.

Such as comparing 2 items.

The bars in a bar graph,

typically are of the same width

but bare no relevance

to the area that they occupy.

In contrast to it,

in a histogram

the dimensions of the bars

are very crucial.

The area of the bar

is directly proportional

to its frequency.

Consequently, so

the width of the class intervals

must be taken into account

whenever you're attempting

to answer relevent questions.

Whenever the width of the

class intervals is non-uniform,

use the minimum class interval

as a standard

and use cross multiplication

to get an accurate representation

of data on the graph.

When it comes to frequency polygons,

the only thing that

you need to do differently

from a histogram,

is to mark the

class mark on the graph.

Class mark is the midpoint

of all the class intervals.

Instead of an entire bar,

you only make one mark.

Then you connect

all of these dots

and get a line.

To make frequency polygon

out of this,

you need to

close the figure.

To do this, add a class

before the first

and after the last classes

with the same width

as the width of the first

and the last classes respectively.