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[Instructor] So this is
a really cool online app

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that helps us see just how powerful

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the central limit theorem can be.

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So we start in this top graph here

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and we can choose the parent population

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that we want to look at.

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We could choose a normal distribution,

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or a skewed distribution,

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or we can create our own distribution.

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And that's what we're gonna do

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to create something completely wacky

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to see how the central
limit theorem will work.

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So let's just draw a distribution
that's totally crazy.

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It's something like that,
and that's a big spike here,

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we go down here, another spike,

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and some more data like this.

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So that's a very non-normal distribution.

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Now, if we want to draw a
sample from this distribution

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of, say, N=5, we draw five
pieces of sample data,

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and from that, we have a sample mean.

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So we do this again,
five new pieces of data,

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and from that, we create another mean.

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We do that a bunch more times.

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So say another 5, 10, 15 times.

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Say we do it 10,000 times,

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now we get a distribution
of the sample means

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that looks like this.

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And if we fit a normal curve,

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we can see that it's
getting really very normal.

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If we go over to the sample statistics,

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we can see that the mean

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of our original distribution was 15.46,

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and the mean of our distribution
of the sampling means

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is also 15.4 something, so
it's getting pretty close.

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We take out our calculators

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and we look at the standard deviation.

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Our original standard deviation is 9.89,

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and if we divide that by
the square root of five,

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we get a 4.42 as a standard error

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of our sampling distribution,

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and that's really close
to what we have here.

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So even though we've got
this crazy wacky distribution

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to start out with, the
distribution of the sampling means

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is indeed approximately normal

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with a mean the same
as our population mean

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and the standard error

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the same as the population
standard deviation

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divided by the square
root of the sample size.

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So what if we change the sample size?

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So say instead of taking samples of N=5,

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we take samples of N=25.

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So now, if we go through

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and now we're pulling 25 pieces of data

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from our original sampling distribution

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and we get a mean from
these data, there it is,

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say we do that five more times,
another five, another five,

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and now we do it 10,000
times like we did before,

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we see that we get another
almost normal distribution

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with the distribution
of the sampling means

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the same mean as the original,

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and the standard error
of the sampling means

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the same as the standard deviation

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from the population divided
by the square root of now 25.

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So let's do that.

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So 9.89 divided by the
square root of 25 is 1.98,

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which is really close to what we see here.

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So you can see that, again,
we have a normal distribution,

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but it's much tighter around the mean.

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So as we increase the size of our sample,

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we get a much more tightly
clustered estimate,

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a much more precise estimate

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of the distribution of the sampling means.