- In this screen cast
you're gonna learn all about the Normal distribution,
also known as the Gaussian distribution.
The Normal distribution is the most widely used model
for the distribution of a continuous random variable.
It oftentimes fits real world data well,
but keep in mind that there is real world data
out there that doesn't necessarily have to fit
the Normal distribution.
There's a lot of different types of distributions.
Histograms often have the shape of the Normal distribution.
It's an approximation
that models real world quantitative phenomena quite well.
It's also known as the Gaussian distribution
or the bell-shaped curve.
Human height, for example,
follows the Normal distribution very well.
The density function
for the Normal distribution is given here.
It's very complex, complicated.
I definitely do not expect you guys to memorize this.
We have something known as the mean, which is mu,
and sigma which is the standard deviation.
I'll talk about the standard Normal distribution
in a follow up screen cast, but it's a special case
of the Normal distribution
where the mean is zero and the standard deviation is one.
The standard Normal density function used
to be used quite a bit
before we had sophisticated software tools like Excel
and MATLAB and Mathcad and Mathematica,
but it is still useful in some scenarios.
We are rarely interested in F of X
for continuous distributions.
The exception is when we want to visualize the distribution
or prepare a plot and I'll show you how to do this
in Excel in a moment.
Instead, we're oftentimes interested
in the cumulative function.
Recall probabilities for continuous variables are obtained
by integrating F of X.
If we wanted to calculate the probability the X lies
between two values A and B,
we would just integrate little F of X from A to B.
There is no analytical solution to the following integral.
So this integral down here, there's no closed form
nice expression for capital F of X.
Instead, what we can do is
we can calculate F of X numerically.
So luckily the integral can be approximated quite accurately
by numerical calculation
and we can also use Excel's norm.dist function.
So instead of integrating,
we can always use the cumulative distributions,
also known as capital F of X.
Recall that the probability that X lies between A and B,
if we know the cumulative distributions capital F,
we can just subtract capital F of A from capital F of B.
In other words, if you wanted to calculate the probability
that the random variable lies between A and B
you would calculate the cumulative capital F
all the way up to B,
that would be capital F of B,
and then you would subtract that first little bit,
which is capital F of A.
So the difference between that would leave you just
with the probability that you are interested in calculating.
In fact, using the differences
in cumulative distributions is the most common
and easy way to determine probabilities related not only
to the Normal distribution,
but all sorts of other continuous probability distributions.
For the Normal distribution, there's a norm.dist function
in Excel and it's very helpful
for working with Normally distributed random variables.
If you don't have access to Excel on a day to day basis,
you're gonna have to standardize everything
and convert it to the standard Normal distribution,
and use tables in the back of your book.
Let's go through an example.
X is a Normally distributed variable, mean equals 10,
standard deviation equals four.
What is the probability that X will be less than six?
This is our distribution.
This is F of X.
What we're interested
in doing is determining the probability that X lies
underneath the curve less than six.
So really we could go to really small values, in fact,
theoretically down to negative infinity
and we need to calculate
or integrate the area underneath that curve.
One way to do this is to numerically integrate
using a computing tool.
Here I think this is Mathcad or Mathematica,
and you can have it numerically integrate
and we get about a 16% chance.
That means 16% of the distribution
of the area lies less than six.
Another way to do this is to use Excel's norm.dist function,
in a little bit I'll show you some more examples
of using this,
but there's a cumulative form,
you always want to use true as the last argument
in the norm.dist function.
If you're trying to calculate cumulative distributions,
the first argument is the X value, six in this case.
Then you have your average of your distribution
and then your standard deviation.
So we had mu and sigma that have been named, and again,
I'll show you how to do this in a minute.
Let's work through a follow up example.
Same scenario, what is the probability that X lies
between eight and 16?
So in other words, what we're trying to do is
we're trying to calculate the area underneath this curve.
Remember, if we're talking
about a probability density function, the total area is one,
and so the integral of F of X DX
between eight and 16 here would be the probability
that X lies between eight and 16.
Let's go ahead and do this in Excel.
Before we solve this,
let me just show you how I created this plot.
And again, you're rarely interested
in just little F of X unless you're making a plot.
For example in Excel.
Here I've named mu cell B1, sigma cell B2,
and I just typed in norm.dist of our X values.
Here I'm creating a bunch of X values
between zero and 20, mu, sigma and false for non cumulative.
And then you can plot this in a nice line graph.
So what we're trying to do is
determine the probability that X lies between eight and 16.
A good way to do this again is
to take the cumulative distribution up to 16.
That would be that entire region.
And then we're gonna subtract out the region
that's up to eight, so we're gonna subtract out that.
So this is F of eight,
and the entire thing that I initially drew was F of 16.
So if we take F of 16 minus F of eight,
then that's going to be the solution.
So that's exactly what we're going to do.
We're going to take norm.dist
of 16 comma mu comma sigma comma true,
that's a cumulative up to 16,
and we're gonna subtract the same thing for up to eight.
And when we do that,
we end up with there's a 62.4% chance
that X will lie between eight and 16.
So that's how we can calculate this probability
with this shaded region.
Let's go through another couple of probabilities,
because these are somewhat important.
Part A, probability that X lies
within one standard deviation of the mean.
Oftentimes this is important to know.
What about two standard deviations of the mean,
and three standard deviations of the mean.
First, let's do this
for one standard deviation from the mean.
So I've got this big long formula,
I'm taking F of mu plus sigma and subtracting
capital F of mu minus sigma.
And in that case we get 68.2.
So there's about 68.2% that the distribution lies
within one standard deviation of the mean.
I'm gonna go ahead and copy this formula
and paste it in these two.
Let's look at two standard deviations,
mu plus two times sigma,
mu minus two times sigma.
That means 95.5% of the distribution lies
within two standard deviations of the mean.
And finally, let's do this for three standard deviations.
So I can replace this with three standard deviations,
and that means about 99.7% of the population lies
plus or minus within three standard deviations of the mean.
And some of you have probably already seen this
in another course or somewhere else.
68% lies within one standard deviation of the mean,
95 within two standard deviations,
and about 99.7 within three standard deviations of the mean.
So hopefully you learned a little bit more
about the Normal distribution in this screen cast.
Thanks for watching.