Original subtitles

Instructor: Welcome back.

In this lecture, we are going to introduce

one of the most commonly found continuous distributions,

the normal distribution.

For starters,

we define a normal distribution using a capital letter N

followed by the mean and variance of the distribution.

We read the following notation

as variable X follows a normal distribution

with mean, mu, and variance, sigma squared.

When dealing with actual data,

we would usually know the numerical values of mu

and sigma squared.

The normal distribution frequently appears in nature,

as well as in life, in various shapes and forms.

For example, the size of a fully grown male lion

follows a normal distribution.

Many records suggest that the average lion weighs

between 150 and 250 kilograms, or 330 to 550 pounds.

Of course, specimens exist which fall outside of this range.

However, lions weighing less than 150

or more than 250 kilograms tend to be the exception

rather than the rule.

Such individuals serve as outliers in our set,

and the more data we gather,

the lower part of the data they represent.

Now that you know what types

of events follow a normal distribution,

let us examine some of its distinct characteristics.

For starters, the graph

of a normal distribution is bell shaped.

Therefore, the majority of the data

is centered around the mean.

Thus, values further away

from the mean are less likely to occur.

Furthermore, we can see

that the graph is symmetric with regards to the mean.

That suggests values equally far away

in opposing directions would still be equally likely.

Let's go back to the lion example from earlier.

If the mean is 400,

symmetry suggests a lion is equally likely

to weigh 350 pounds and 450 pounds,

since both are 50 pounds away from the mean.

All right, for anybody interested,

you can find the CDF and the PDF of the normal distribution

in the additional materials for this lecture.

Instead of going

through the complex algebraic simplifications

in this lecture, we are simply going to talk

about the expected value and the variance.

The expected value for a normal distribution

equals its mean, mu, whereas its variance, sigma squared,

is usually given when we define the distribution.

However, if it isn't,

we can deduce it from the expected value.

To do so, we must apply the formula we showed earlier.

The variance of a variable is equal

to the expected value of the squared variable

minus the squared expected value of the variable.

Good job.

Another peculiarity of the normal distribution

is the 68, 95, 99.7 law.

This law suggests that for any normally distributed event,

68% of all outcomes fall within one standard deviation

away from the mean,

95% fall within two standard deviations,

and 99.7 within three.

The last part really emphasizes the fact

that outliers are extremely rare in normal distributions.

It also suggests how much we know about a data set

if we only have the information

that it is normally distributed.

Fantastic work, everyone.

Before we move on to other types of distributions,

you need to know

that we can analyze any normal distribution.

To do this, we need to standardize the distribution,

which we will explain in detail in the next video.

Thanks for watching.