Original subtitles

Instructor: Welcome back everybody.

Towards the end of the last lecture,

we mentioned standardizing,

but didn't explain what it is and why we use it.

Before we understand this concept

we need to explain what a transformation is.

So a transformation is a way in which we can alter

every element of a distribution to get a new distribution

with similar characteristics.

For normal distributions we can use addition,

subtraction, multiplication, and division,

without changing the type of the distribution.

For instance, if we add a constant

to every element of a normal distribution

the new distribution would still be normal.

Let's discuss the four algebraic options

and see how each one affects the graph.

If we had a constant like three to the entire distribution

then we simply need to move the graph three places

to the right.

Similarly, if we subtract a number from every element

we would simply move our current graph

to the left to get the new one.

If we multiply the function by a constant

it will shrink that many times,

and if we divide every element by a number,

the graph will expand.

However, if we multiply or divide by a number

between zero and one, the opposing effects will occur.

For example, dividing by a half is the same as multiplying

by two, so the graph will shrink

even though we are dividing.

All right.

Now that you know what a transformation is,

we can explain standardizing.

Standardizing is a special kind of transformation

in which we make the expected value equal to zero

and the variance equal to one.

The distribution we get after standardizing

any normal distribution is called

a standard normal distribution.

In addition to the 68, 95, 99.7 rule,

a table exists which summarizes the most commonly

used values for the CDF of a standard normal distribution.

This table is known as the standard normal distribution

table or the Z-score table.

Okay, so far we have learned what standardizing is

and why it's convenient.

What we haven't talked about is how to do it.

First, we wish to move the graph either to the left

or to the right until it's mean equals zero.

The way we would do that is by subtracting the mean Mu,

from every element.

After this, to make the standardization complete

we need to make sure the standard deviation is one.

To do so, we would have to divide every element

of the newly obtained distribution

by the value of the standard deviation, sigma.

If we denote the standard normal distribution with Z,

then for any normally distributed variable Y,

Z equals Y minus Mu over sigma.

This equation expresses the transformation we

use when standardizing.

Amazing.

Applying this single transformation

for any normal distribution would result in a standard

normal distribution, which is convenient.

Essentially, every element of the non-standardized

distribution is represented in the new distribution

by the number of standard deviations it is

away from the mean.

For instance, if a value Y is 2.3 standard deviations

away from the mean, it's equivalent value Z

would be equal to 2.3.

Standardizing is incredibly useful when

we have a normal distribution.

However, we cannot always anticipate

the data is spread out that way.

A crucial fact to remember about the normal distribution

is that it requires a lot of data.

If our sample is limited, we run the risk of outliers

drastically affecting our analysis.

In cases where we have less than 30 entries,

we usually avoid assuming a normal distribution.

However, there is a small sample size approximation

of a normal distribution called

the student's T distribution.

And we are going to focus on that in our next lecture.

Thanks for watching.