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We've seen in the last several
videos you start off with

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any crazy distribution.

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It doesn't have to be
crazy, it could be a nice

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normal distribution.

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But to really make the point
that you don't have to have

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a normal distribution I
like to use crazy ones.

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So let's say you have some kind
of crazy distribution that

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looks something like that.

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It could look like anything.

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So we've seen multiple times
you take samples from

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this crazy distribution.

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So let's say you were to take
samples of n is equal to 10.

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So we take 10 instances of this
random variable, average them

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out, and then plot our average.

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We plot our average.

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We get 1 instance there.

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We keep doing that.

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We do that again.

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We take 10 samples from this
random variable, average

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them, plot them again.

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You plot again and eventually
you do this a gazillion times--

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in theory an infinite number of
times-- and you're going to

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approach the sampling
distribution of the sample

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mean. n equal 10 is not going
to be a perfect normal

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distribution but it's
going to be close.

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It'd be perfect only
if n was infinity.

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But let's say we eventually--
all of our samples we get a lot

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of averages that are there that
stacks up, that stacks up

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there, and eventually will
approach something that

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looks something like that.

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And we've seen from the last
video that one-- if let's say

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we were to do it again and this
time let's say that n is equal

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to 20-- one, the distribution
that we get is going

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to be more normal.

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And maybe in future videos
we'll delve even deeper into

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things like kurtosis and skew.

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But it's going to
be more normal.

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But even more important here or
I guess even more obviously

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to us, we saw that in the
experiment it's going to have

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a lower standard deviation.

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So they're all going to
have the same mean.

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Let's say the mean here is,
I don't know, let's say

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the mean here is 5.

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Then the mean here is
also going to be 5.

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The mean of our sampling
distribution of the sample

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mean is going to be 5.

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It doesn't matter
what our n is.

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If our n is 20 it's
still going to be 5.

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But our standard deviation
is going to be less than

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either of these scenarios.

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And we saw that just
by experimenting.

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It might look like this.

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It's going to be more normal
but it's going to have a

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tighter standard deviation.

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So maybe it'll look like that.

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And if we did it with an even
larger sample size-- let me do

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that in a different color-- if
we did that with an even larger

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sample size, n is equal to 100,
what we're going to get is

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something that fits the normal
distribution even better.

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We take a hundred instances
of this random variable,

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average them, plot it.

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A hundred instances of
this random variable,

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average them, plot it.

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And we just keep doing that.

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If we keep doing that, what
we're going to have is

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something that's even more
normal than either of these.

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So it's going to be a much
closer fit to a true

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normal distribution.

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But even more obvious to
the human, it's going

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to be even tighter.

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So it's going to be a very
low standard deviation.

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It's going to look
something like that.

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And I'll show you on the
simulation app in the next or

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probably later in this video.

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So two things happen.

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As you increase your sample
size for every time you

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do the average, two
things are happening.

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You're becoming more normal
and your standard deviation

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is getting smaller.

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So the question might
arise is there a formula?

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So if I know the standard
deviation-- so this is my

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standard deviation of just my
original probability density

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function, this is the mean of
my original probability

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density function.

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So if I know the standard
deviation and I know n-- n is

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going to change depending on
how many samples I'm taking

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every time I do a sample mean--
if I know that my standard

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deviation, or maybe if I
know my variance, right?

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The variance to just the
standard deviation squared.

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If you don't remember
that you might want to

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review those videos.

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But if I know the variance of
my original distribution and if

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I know what my n is-- how many
samples I'm going to take every

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time before I average them in
order to plot one thing in my

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sampling distribution of my
sample mean-- is there a way to

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predict what the mean of
these distributions are?

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And so-- I'm sorry, the
standard deviation of

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these distributions.

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And so you don't get confused
between that and that,

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let me say the variance.

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If you know the variance
you can figure out the

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standard deviation.

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One is just the square
root of the other.

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So this is the variance of
our original distribution.

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Now to show that this is the
variance of our sampling

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distribution of our sample mean
we'll write it right here.

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This is the variance of our
mean of our sample mean.

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Remember the sample--
our true mean is this.

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The Greek letter Mu
is our true mean.

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This is equal to the mean,
while an x a line over

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it means sample mean.

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So here what we're saying is
this is the variance of our

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sample mean, that this is going
to be true distribution.

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This isn't an estimate.

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There's some-- you know, if we
magically knew distribution--

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there's some true
variance here.

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And of course the mean-- so
this has a mean-- this right

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here, we can just get our
notation right, this is the

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mean of the sampling
distribution of the

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sampling mean.

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So this is the mean
of our means.

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It just happens to
be the same thing.

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This is the mean of
our sample means.

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It's going to be the same thing
as that, especially if we do

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the trial over and over again.

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But anyway, the point of this
video, is there any way to

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figure out this variance given
the variance of the original

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distribution and your n?

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And it turns out there is.

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And I'm not going to
do a proof here.

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I really want to give you
the intuition of it.

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I think you already do have the
sense that every trial you

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take-- if you take a hundred,
you're much more likely when

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you average those out, to get
close to the true mean than if

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you took an n of
2 or an n of 5.

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You're just very unlikely to be
far away, right, if you took

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100 trials as opposed
to taking 5.

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So I think you know that
in some way it should be

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inversely proportional to n.

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The larger your n the smaller
a standard deviation.

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And actually it turns out it's
about as simple as possible.

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It's one of those magical
things about mathematics.

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And I'll prove it
to you one day.

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I want to give you
working knowledge first.

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In statistics, I'm always
struggling whether I should be

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formal in giving you rigorous
proofs but I've kind of come to

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the conclusion that it's more
important to get the working

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knowledge first in statistics
and then later, once you've

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gotten all of that down, we can
get into the real deep math

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of it and prove it to you.

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But I think experimental proofs
are kind of all you need for

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right now, using those
simulations to show that

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they're really true.

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So it turns out that the
variance of your sampling

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distribution of your sample
mean is equal to the

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variance of your original
distribution-- that guy

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right there-- divided by n.

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That's all it is.

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So if this up here has a
variance of-- let's say this up

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here has a variance of 20-- I'm
just making that number up--

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then let's say your n is 20.

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Then the variance of your
sampling distribution of your

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sample mean for an n of 20,
well you're just going to take

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that, the variance up here--
your variance is 20--

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divided by your n, 20.

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So here your variance is
going to be 20 divided by

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20 which is equal to 1.

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This is the variance of
your original probability

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distribution and
this is your n.

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What's your standard
deviation going to be?

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What's going to be the
square root of that, right?

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Standard deviation is going
to be square root of 1.

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Well that's also going to be 1.

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So we could also write this.

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We could take the square root
of both sides of this and say

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the standard deviation of the
sampling distribution

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standard-- the standard
deviation of the sampling

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distribution of the sample mean
is often called the standard

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deviation of the mean.

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And it's also called-- I'm
going to write this down-- the

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standard error of the mean.

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All of these things that I just
mentioned, they all just mean

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the standard deviation of the
sampling distribution

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of the sample mean.

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That's why this is confusing
because you use the word mean

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and sample over and over again.

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And if it confuses
you let me know.

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I'll do another video or pause
and repeat or whatever.

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But if we just take the square
root of both sides, the

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standard error of the mean or
the standard deviation of the

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sampling distribution of the
sample mean is equal to the

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standard deviation of your
original function-- of your

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original probability density
function-- which could be very

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non-normal, divided by
the square root of n.

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I just took the square root of
both sides of this equation.

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I personally like to remember
this: that the variance is just

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inversely proportional to n.

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And then I like to
go back to this.

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Because this is very
simple in my head.

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You just take the
variance, divide it by n.

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Oh and if I want the standard
deviation, I just take the

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square roots of both sides
and I get this formula.

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So here the standard
deviation-- when n is 20-- the

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standard deviation of the
sampling distribution of the

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sample mean is going to be 1.

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Here when n is 100, our
variance here when

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n is equal to 100.

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So our variance of the sampling
mean of the sample distribution

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or our variance of the mean--
of the sample mean, we

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could say-- is going to be
equal to 20-- this guy's

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variance-- divided by n.

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So it equals-- n is
100-- so it equals 1/5.

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Now this guy's standard
deviation or the standard

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deviation of the sampling
distribution of the sample mean

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or the standard error of the
mean is going to be the

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square root of that.

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So 1 over the square root of 5.

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And so this guy's will be a
little bit under 1/2 the

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standard deviation while
this guy had a standard

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deviation of 1.

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So you see, it's
definitely thinner.

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Now I know what you're saying.

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Well, Sal, you just gave
a formula, I don't

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necessarily believe you.

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Well let's see if we can
prove it to ourselves

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using the simulation.

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So just for fun let me make
a-- I'll just mess with this

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distribution a little bit.

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So that's my new distribution.

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And let me take an n of-- let
me take two things that's easy

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to take the square root of
because we're looking at

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standard deviations.

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So we take an n of
16 and an n of 25.

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Let's do 10,000 trials.

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So in this case every one of
the trials we're going to take

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16 samples from here, average
them, plot it here, and

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then do a frequency plot.

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Here we're going to do 25 at a
time and then average them.

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I'll do it once animated
just to remember.

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So I'm taking 16
samples, plot it there.

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I take 16 samples as described
by this probability density

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function-- or 25 now,
plot it down here.

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Now if I do that 10,000
times, what do I get?

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00:10:03,420 --> 00:10:06,710
All right, so here, just
visually you can tell just when

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00:10:06,710 --> 00:10:08,810
n was larger, the standard
deviation here is smaller.

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00:10:08,810 --> 00:10:10,020
This is more squeezed together.

250
00:10:10,020 --> 00:10:12,490
But actually let's
write this stuff down.

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00:10:12,490 --> 00:10:14,420
Let's see if I can
remember it here.

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00:10:14,420 --> 00:10:17,370
So in this random distribution
I made my standard

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00:10:17,370 --> 00:10:19,160
deviation was 9.3.

254
00:10:19,160 --> 00:10:20,460
I'm going to remember these.

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00:10:20,460 --> 00:10:24,470
Our standard deviation for
the original thing was 9.3.

256
00:10:24,470 --> 00:10:27,980
And so standard deviation here
was 2.3 and the standard

257
00:10:27,980 --> 00:10:29,590
deviation here is 1.87.

258
00:10:29,590 --> 00:10:33,290
Let's see if it conforms
to our formula.

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00:10:33,290 --> 00:10:35,240
So I'm going to take this off
screen for a second and I'm

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00:10:35,240 --> 00:10:38,900
going to go back and
do some mathematics.

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00:10:38,900 --> 00:10:41,030
So I have this on my
other screen so I can

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00:10:41,030 --> 00:10:42,750
remember those numbers.

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00:10:42,750 --> 00:10:47,220
So in the trial we just did,
my wacky distribution had a

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00:10:47,220 --> 00:10:52,650
standard deviation of 9.3.

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00:10:52,650 --> 00:10:57,680
When n is equal to-- let me do
this in another color-- when n

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00:10:57,680 --> 00:11:01,900
was equal to 16, just doing the
experiment, doing a bunch of

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00:11:01,900 --> 00:11:04,460
trials and averaging and doing
all the things, we got the

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00:11:04,460 --> 00:11:08,070
standard deviation of the
sampling distribution of the

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00:11:08,070 --> 00:11:10,380
sample mean or the standard
error of the mean, we

270
00:11:10,380 --> 00:11:15,570
experimentally determined
it to be 2.33.

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00:11:15,570 --> 00:11:21,500
And then when n is equal to 25
we got the standard error of

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00:11:21,500 --> 00:11:24,900
the mean being equal to 1.87.

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00:11:24,900 --> 00:11:28,330
Let's see if it conforms
to our formulas.

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00:11:28,330 --> 00:11:32,680
So we know that the variance or
we could almost say the

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00:11:32,680 --> 00:11:36,010
variance of the mean or the
standard error-- the variance

276
00:11:36,010 --> 00:11:39,230
of the sampling distribution of
the sample mean is equal to the

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00:11:39,230 --> 00:11:42,050
variance of our original
distribution divided by n, take

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00:11:42,050 --> 00:11:45,490
the square roots of both sides,
and then you get the standard

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00:11:45,490 --> 00:11:48,770
error of the mean is equal to
the standard deviation of your

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00:11:48,770 --> 00:11:51,800
original distribution divided
by the square root of n.

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00:11:51,800 --> 00:11:54,350
So let's see if this works
out for these two things.

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00:11:54,350 --> 00:11:58,670
So if I were to take 9.3--
so let me do this case.

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00:11:58,670 --> 00:12:03,870
So 9.3 divided by the
square root of 16, right?

284
00:12:03,870 --> 00:12:05,150
N is 16.

285
00:12:05,150 --> 00:12:07,050
So divided by the square
root of 16, which is

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00:12:07,050 --> 00:12:09,300
4, what do I get?

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00:12:09,300 --> 00:12:11,566
So 9.3 divided by 4.

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00:12:11,566 --> 00:12:14,840
Let me get a little
calculator out here.

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00:12:14,840 --> 00:12:15,550
Let's see.

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00:12:15,550 --> 00:12:18,720
We have-- let me clear it
out-- we want to divide

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00:12:18,720 --> 00:12:21,000
9.3 divided by 4.

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00:12:21,000 --> 00:12:24,930
9.3 three divided by our
square root of n. n was 16.

293
00:12:24,930 --> 00:12:32,060
So divided by 4 is
equal to 2.32.

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00:12:32,060 --> 00:12:41,540
So this is equal to 2.32 which
is pretty darn close to 2.33.

295
00:12:41,540 --> 00:12:43,160
This was after 10,000 trials.

296
00:12:43,160 --> 00:12:45,780
Maybe right after this I'll see
what happens if we did 20,000

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00:12:45,780 --> 00:12:48,960
or 30,000 trials where we take
samples of 16 and average them.

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00:12:48,960 --> 00:12:50,340
Now let's look at this.

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00:12:50,340 --> 00:12:55,400
Here we would take 9.3-- so let
me draw a little line here.

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00:12:55,400 --> 00:12:57,300
Let me scroll over,
that might be better.

301
00:12:57,300 --> 00:13:00,340
So we take our standard
deviation of our

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00:13:00,340 --> 00:13:01,790
original distribution.

303
00:13:01,790 --> 00:13:05,280
So just that formula that we've
derived right here would tell

304
00:13:05,280 --> 00:13:09,180
us that our standard error
should be equal to the standard

305
00:13:09,180 --> 00:13:13,350
deviation of our original
distribution, 9.3, divided by

306
00:13:13,350 --> 00:13:15,400
the square root of n, divided
by the square root

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00:13:15,400 --> 00:13:16,380
of 25, right?

308
00:13:16,380 --> 00:13:18,410
4 was just the
square root of 16.

309
00:13:18,410 --> 00:13:21,860
So this is equal to
9.3 divided by 5.

310
00:13:21,860 --> 00:13:23,820
And let's see if it's 1.87.

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00:13:23,820 --> 00:13:28,320
So let me get my
calculator back.

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00:13:28,320 --> 00:13:36,410
So if I take 9.3 divided
by 5, what do I get?

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00:13:36,410 --> 00:13:41,720
1.86 which is very
close to 1.87.

314
00:13:41,720 --> 00:13:49,480
So we got in this case 1.86.

315
00:13:49,480 --> 00:13:53,150
So as you can see what we got
experimentally was almost

316
00:13:53,150 --> 00:13:56,010
exactly-- and this was after
10,000 trials-- of what

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00:13:56,010 --> 00:13:56,600
you would expect.

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00:13:56,600 --> 00:13:58,690
Let's do another 10,000.

319
00:13:58,690 --> 00:14:00,220
So you've got another
10,000 trials.

320
00:14:00,220 --> 00:14:01,550
Well we're still
in the ballpark.

321
00:14:01,550 --> 00:14:04,920
We're not going to-- maybe I
can't hope to get the exact

322
00:14:04,920 --> 00:14:07,350
number rounded or whatever.

323
00:14:07,350 --> 00:14:10,690
But as you can see, hopefully
that'll be pretty satisfying to

324
00:14:10,690 --> 00:14:14,410
you, that the variance of the
sampling distribution of the

325
00:14:14,410 --> 00:14:21,500
sample mean is just going to be
equal to the variance of your

326
00:14:21,500 --> 00:14:23,790
original distribution, no
matter how wacky that

327
00:14:23,790 --> 00:14:27,370
distribution might be, divided
by your sample size-- by the

328
00:14:27,370 --> 00:14:33,530
number of samples you take for
every basket that you average I

329
00:14:33,530 --> 00:14:35,220
guess is the best way
to think about it.

330
00:14:35,220 --> 00:14:37,710
You know, sometimes this can
get confusing because you are

331
00:14:37,710 --> 00:14:40,275
taking samples of averages
based on samples.

332
00:14:40,275 --> 00:14:43,300
So when someone says sample
size, you're like, is sample

333
00:14:43,300 --> 00:14:46,510
size the number of times I
took averages or the number

334
00:14:46,510 --> 00:14:48,780
of things I'm taking
averages of each time?

335
00:14:48,780 --> 00:14:51,160
And you know, it doesn't
hurt to clarify that.

336
00:14:51,160 --> 00:14:52,790
Normally when they talk
about sample size

337
00:14:52,790 --> 00:14:54,220
they're talking about n.

338
00:14:54,220 --> 00:14:57,570
And, at least in my head, when
I think of the trials as you

339
00:14:57,570 --> 00:15:00,700
take a sample size of 16, you
average it, that's the one

340
00:15:00,700 --> 00:15:01,990
trial, and then you plot it.

341
00:15:01,990 --> 00:15:03,680
Then you do it again and
you do another trial.

342
00:15:03,680 --> 00:15:04,980
And you do it over
and over again.

343
00:15:04,980 --> 00:15:06,930
But anyway, hopefully this
makes everything clear and then

344
00:15:06,930 --> 00:15:11,340
you now also understand how to
get to the standard

345
00:15:11,340 --> 00:15:13,750
error of the mean.

346
00:15:13,750 --> 00:15:14,848