This is Stephanie from StatisticsHowTo.com, and in this video I am going to be showing
you about the variance of the sampling distribution of the sample mean.
I am going to work through a sample problem to find the variance of the sampling distribution
of the sample mean. And I will also be showing you why it works
and what exactly the variance means when you are referring to the sampling distribution
of the sample mean. The formula for the variance of the sampling
distribution of the sampling mean. And we write it like this with a sub M.
This is the variance of the sampling distribution of the sample mean is equal to the population
variance. That is just the standard deviation squared
divided by the number of items in your sample. Let us say we had a population with standard
deviation of say 20. And a sample size n of 19.
We can use this formula up here to figure out our variance of the sampling distribution
of the sample mean. This is going to equal our standard deviation
squared. 20 squared all over n, our sample size is
19. Which is going to equal 400 divided by 19,
which is equal to about 21.05. This is our variance and if you have studied
variance in statistics, this number does not actually give you a concrete
piece of information about much, but a variance of 21 means that there is variance
in the means. The larger the sample size the closer these variance
will get to 0. And that means that your probability distribution
of the means is actually a representative of the actual population.
Let me show you exactly what I mean by that. This is a probability distribution for the
sample mean. And as you can see, there is a couple hundred
samples that have been drawn each of these crosses marks one sample.
And the mean for the population is 6 And when we draw our sample means and we find
out what the average is the bulk of the averages from our samples
are going to be around six. But as you can see thiere is, quite a lot of variance
here. It could be just above six or it could be
just below six. The idea is we want to continue to draw samples
as many as possible. Central limit theorem tells us that the more
the samples we draw, the closer we are to the actual mean in
our sampling distribution. The larger the sample size, the smaller the
variance. Now why is that?
Let us take a look at the formula again. In the formula, we are dividing by the sample
size, so the larger these denominator gets,
the smaller the variance for the sampling distribution will get.
Let show you how this works with a quick example we calculated just a second ago.
That if we had a sample size of 19, then 400 divided by 19 is 21.05.
What if we increase the sample size, what is going to happen to our variance?
If we increase our sample size to 400, our variance is going to equal 1,
which is significantly smaller than 21.05. If we went all the way up to say 10,000,
so then we have got 400 divided by 10,000. Our variance is going to be 0.04.
In other words if we draw a 10,000 samples. Our variance is small and our sample mean
is going to be extremely close to the actual mean for the population.
I am going to show you one more example of how to find that variance of the sampling
distribution of the sample mean. Your population standard deviation is 290.
And you are drawing a sample size of a thousand. You want to find out what is the variance
of the sampling distribution of the sample mean. What are these going to be your standard deviation
squared. 290 squared divided by your sample size of
a thousand. That equals 84.1
If we increase the sample size to 10,000, our variance is going to drop to 8.41.
That is how to calculate the variance of the sampling distribution of the sample mean.
Visit us at StatisticsHowTo.com for more tips and videos on Elementary statistics and AP
statistics.