Welcome to a lesson
on the Central Limit Theorem for Sample Means.
The central limit theorem or CLT for short
is one of the most powerful and useful ideas
in all of statistics.
There are two alternative forms of the theorem
and both alternatives are concerned
with drawing finite samples size n
from a population with a known mean Mu
and a known standard deviation Sigma.
The first alternative says
that if we collect samples of size n
with a large enough n,
calculate each sample's mean
and create a histogram of those means,
the resulting histogram will tend to
an approximate normal bell shaped curve
or a normal distribution.
The second alternative says
that if we again collect samples of size n
that are large enough,
calculate the sum of each sample and create a histogram,
then the resulting histogram will again,
tend to a bell shaped curve or a normal distribution.
The size of the sample n that is required
in order to be large enough,
depends on the original population
from which the samples are drawn.
The sample size should be at least 30
or the data should come from a normal distribution.
If the original population is far from normal,
then more observations are needed
for the sample of means or sums to be normal.
Sampling is done with replacement.
It will be difficult to overstate the importance
of the central limit theorem in statistical theory.
Knowing that data, even if its distribution is not normal,
behave in a predictable way is a powerful tool.
Suppose x is a random variable
with a distribution that may be known or unknown.
Using a subscript that matches the random variable,
suppose Mu sub x equals the mean of X
and Sigma sub x is equal to the standard deviation of X.
If you draw random samples of size n, as n increases,
the random variable x-bar,
which consists of sample means
tends to be normally distributed.
Indicated using this notation here
where the mean of X-bar
is Mu sub x,
and the standard deviation of X-bar
is equal to the standard deviation of x
divided by the square root of n.
Where again, Mu sub x is the average or mean
of both x and x-bar.
And Sigma sub x-bar again is equal to
Sigma sub x, the standard deviation of X
divided by the square root of n,
where n is the sample size.
And Sigma sub x-bar
is called the standard error of the mean.
Also notice how as n increases,
the denominator of Sigma sub x-bar increases
and therefore Sigma sub x-bar decreases.
The central theorem for sample means says,
if you keep drawing larger and larger samples
and calculating their means,
the sample means form their own normal distribution,
which is called the sampling distribution.
The normal distribution has the same mean
as the original distribution
and a variance that equals the original variance
divided by the sample size.
The standard deviation is the square root of the variance,
so the standard deviation of the sampling distribution
is the standard deviation of original distribution
divided by the square root of n,
given by the notation shown here at the top.
The variable n is the number of values
that are averaged together,
not the number of times the experiment is done.
To put it more formerly,
if you draw random samples of size n,
the distribution of the random variable X-bar,
which consists of the sample means
is called the sampling distribution of the mean.
The sampling distribution of the mean
approaches a normal distribution,
as n the sampling size increases.
The random variable X-bar has a different z-score
associated with it from that of the original variable x.
The mean small x-bar is the value of big X-bar
in one sample.
So the z-score is equal to x-bar
minus Mu sub x
divided by the quotient of Sigma sub x
and the square root of n,
which is equivalent to x-bar minus Mu sub x
divided by Sigma sub x-bar.
Remember Sigma sub x-bar is equal to this quotient here
in the denominator.
To better understand this,
let's look at an animation.
In this animation the first graph is the parent population,
the second graph is the sample data
and the third graph is the distribution of the means.
So if you take more and more samples,
we should begin to see the distribution of the means
approach a normal distribution.
For this example,
let's go ahead and select a sample size of 25.
I know earlier we said,
we should have a sample size of at least 30,
but this animation only goes to a maximum sample size of 25,
which should be enough.
So at the top here we can select a variety of distributions.
We have a normal distribution,
a uniform distribution,
a skewed distribution,
as well as we can make a custom distribution.
And again regardless of the distribution of the population,
as we find the means of each sample
and then plot the distribution of the means,
we should begin to see a normal distribution
for the distribution of means.
So let's find one sample from the parent population
which has a skewed distribution.
So here we see the 20 samples being drawn.
I'm sorry, 25 samples.
And at the bottom we have
the first mean of the first sample.
Let's go and do that again.
So here's the second sample.
And now we'll go ahead and speed this up.
Let's go ahead and find five samples.
Five more.
Let's try 10,000 more,
10,000, more,
10,000 more and so on.
Notice right now we've taken 50,000, 12 samples,
and notice the distribution of the means
does look like a normal distribution or a bell shaped curve,
which we can see here.
And this is the idea behind the central limit theorem.
Also notice how the mean of the parent population
and the mean of the distribution of means
are the same.
The standard deviations are different because
remember to find the standard deviation
for the distribution of the means,
we take the standard deviation of the population
and divide by this square root of n,
which in this case it'd be the square 25 or five.
Now let's look at a different parent population.
Let's try a uniform distribution.
Let's go ahead and find one sample,
and let's go ahead and increase the number of samples
by 10,000 several times.
[clicks mouse]
We can quickly see how the distribution of the means
is approaching a normal distribution or a bell shaped curve,
which is indicated by the central limit theorem.
And I'll do this one more time.
Let's create our own custom distribution
for the parent population,
maybe something like this.
Again regardless of this distribution,
the distribution of the means
should approach a normal distribution.
So let's go in and find one sample of 25.
And now let's increase number of samples
by let's say five at a time.
And we can really see the development
of the normal distribution
for the distribution of the means.
[clicks mouse]
Let's increase it by 10,000 now,
then we can really see how the distribution of the means
is approaching a normal distribution.
And once again notice how the mean of the population
and the mean of the distribution of means
is almost the same.
The standard deviations are different
because remember to find the standard deviation
of the distribution of means,
we take the standard deviation of the population
and divide by the square root of n,
which in this case is a square to 25,
which is equal to five.
So notice how the standard deviation
of the distribution of means is much smaller
than the standard deviation of the population.
The content from this lesson
has been adapted from
the Introductory Statistics Textbook from OpenStax
and the animation of the central limit theorem
is from onlinestatbook.com. and here is a direct link.
I hope you found this helpful.