[Instructor] So this is a really cool online app
that helps us see just how powerful
the central limit theorem can be.
So we start in this top graph here
and we can choose the parent population
that we want to look at.
We could choose a normal distribution,
or a skewed distribution,
or we can create our own distribution.
And that's what we're gonna do
to create something completely wacky
to see how the central limit theorem will work.
So let's just draw a distribution that's totally crazy.
It's something like that, and that's a big spike here,
we go down here, another spike,
and some more data like this.
So that's a very non-normal distribution.
Now, if we want to draw a sample from this distribution
of, say, N=5, we draw five pieces of sample data,
and from that, we have a sample mean.
So we do this again, five new pieces of data,
and from that, we create another mean.
We do that a bunch more times.
So say another 5, 10, 15 times.
Say we do it 10,000 times,
now we get a distribution of the sample means
that looks like this.
And if we fit a normal curve,
we can see that it's getting really very normal.
If we go over to the sample statistics,
we can see that the mean
of our original distribution was 15.46,
and the mean of our distribution of the sampling means
is also 15.4 something, so it's getting pretty close.
We take out our calculators
and we look at the standard deviation.
Our original standard deviation is 9.89,
and if we divide that by the square root of five,
we get a 4.42 as a standard error
of our sampling distribution,
and that's really close to what we have here.
So even though we've got this crazy wacky distribution
to start out with, the distribution of the sampling means
is indeed approximately normal
with a mean the same as our population mean
and the standard error
the same as the population standard deviation
divided by the square root of the sample size.
So what if we change the sample size?
So say instead of taking samples of N=5,
we take samples of N=25.
So now, if we go through
and now we're pulling 25 pieces of data
from our original sampling distribution
and we get a mean from these data, there it is,
say we do that five more times, another five, another five,
and now we do it 10,000 times like we did before,
we see that we get another almost normal distribution
with the distribution of the sampling means
the same mean as the original,
and the standard error of the sampling means
the same as the standard deviation
from the population divided by the square root of now 25.
So let's do that.
So 9.89 divided by the square root of 25 is 1.98,
which is really close to what we see here.
So you can see that, again, we have a normal distribution,
but it's much tighter around the mean.
So as we increase the size of our sample,
we get a much more tightly clustered estimate,
a much more precise estimate
of the distribution of the sampling means.