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Central limit theorem | Inferential statistics | Probability and Statistics | Khan Academy - YouTube
Channel: Khan Academy
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In this video, I want to
talk about what is easily
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one of the most fundamental and
profound concepts in statistics
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and maybe in all of mathematics.
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And that's the
central limit theorem.
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And what it tells us
is we can start off
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with any distribution that
has a well-defined mean and
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variance-- and if it has
a well-defined variance,
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it has a well-defined
standard deviation.
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And it could be a continuous
distribution or a discrete one.
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I'll draw a discrete one,
just because it's easier
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to imagine, at least for
the purposes of this video.
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So let's say I have a discrete
probability distribution
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function.
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And I want to be
very careful not
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to make it look anything close
to a normal distribution.
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Because I want to show you
the power of the central limit
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theorem.
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So let's say I have
a distribution.
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Let's say it could take
on values 1 through 6.
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1, 2, 3, 4, 5, 6.
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It's some kind of crazy dice.
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It's very likely to get a one.
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Let's say it's
impossible-- well,
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let me make that
a straight line.
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You have a very high
likelihood of getting a 1.
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Let's say it's
impossible to get a 2.
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Let's say it's an OK likelihood
of getting a 3 or a 4.
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Let's say it's
impossible to get a 5.
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And let's say it's very
likely to get a 6 like that.
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So that's my probability
distribution function.
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If I were to draw a
mean-- this the symmetric,
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so maybe the mean would
be something like that.
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The mean would be halfway.
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So that would be my
mean right there.
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The standard
deviation maybe would
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look-- it would be
that far and that
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far above and below the mean.
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But that's my discrete
probability distribution
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function.
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Now what I'm going to do
here, instead of just taking
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samples of this
random variable that's
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described by this probability
distribution function,
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I'm going to take samples of it.
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But I'm going to
average the samples
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and then look at
those samples and see
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the frequency of the
averages that I get.
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And when I say average,
I mean the mean.
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Let me define something.
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Let's say my sample size-- and
I could put any number here.
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But let's say first off we try a
sample size of n is equal to 4.
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And what that means is I'm going
to take four samples from this.
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So let's say the first
time I take four samples--
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so my sample sizes is
four-- let's say I get a 1.
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Let's say I get another 1.
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And let's say I get a 3.
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And I get a 6.
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So that right there is my
first sample of sample size 4.
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I know the terminology
can get confusing.
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Because this is the sample
that's made up of four samples.
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But then when we talk about the
sample mean and the sampling
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distribution of the
sample mean, which we're
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going to talk more and more
about over the next few videos,
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normally the sample refers
to the set of samples
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from your distribution.
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And the sample size tells
you how many you actually
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took from your distribution.
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But the terminology
can be very confusing,
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because you could easily view
one of these as a sample.
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But we're taking four
samples from here.
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We have a sample size of four.
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And what I'm going to do is
I'm going to average them.
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So let's say the mean-- I
want to be very careful when
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I say average.
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The mean of this first
sample of size 4 is what?
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1 plus 1 is 2.
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2 plus 3 is 5.
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5 plus 6 is 11.
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11 divided by 4 is 2.75.
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That is my first sample mean
for my first sample of size 4.
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Let me do another one.
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My second sample of size 4,
let's say that I get a 3, a 4.
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Let's say I get another 3.
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And let's say I get a 1.
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I just didn't happen
to get a 6 that time.
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And notice I can't
get a 2 or a 5.
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It's impossible for
this distribution.
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The chance of getting
a 2 or 5 is 0.
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So I can't have any
2s or 5s over here.
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So for the second
sample of sample size 4,
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my second sample mean is
going to be 3 plus 4 is 7.
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7 plus 3 is 10 plus 1 is 11.
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11 divided by 4,
once again, is 2.75.
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Let me do one more,
because I really
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want to make it clear
what we're doing here.
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So I do one more.
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Actually, we're going
to do a gazillion more.
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But let me just do
one more in detail.
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So let's say my third
sample of sample size 4--
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so I'm going to
literally take 4 samples.
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So my sample is
made up of 4 samples
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from this original
crazy distribution.
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Let's say I get a 1,
a 1, and a 6 and a 6.
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And so my third sample mean
is going to be 1 plus 1 is 2.
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2 plus 6 is 8.
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8 plus 6 is 14.
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14 divided by 4 is 3 and 1/2.
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And as I find each
of these sample
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means-- so for each of my
samples of sample size 4,
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I figure out a mean.
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And as I do each
of them, I'm going
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to plot it on a
frequency distribution.
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And this is all going to
amaze you in a few seconds.
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So I plot this all on a
frequency distribution.
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So I say, OK, on
my first sample,
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my first sample mean was 2.75.
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So I'm plotting the actual
frequency of the sample
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means I get for each sample.
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So 2.75, I got it one time.
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So I'll put a little plot there.
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So that's from that
one right there.
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And the next time,
I also got a 2.75.
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That's a 2.75 there.
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So I got it twice.
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So I'll plot the
frequency right there.
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Then I got a 3 and 1/2.
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So all the possible values,
I could have a three,
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I could have a 3.25, I
could have a 3 and 1/2.
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So then I have the 3 and 1/2,
so I'll plot it right there.
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And what I'm going
to do is I'm going
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to keep taking these samples.
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Maybe I'll take 10,000 of them.
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So I'm going to keep
taking these samples.
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So I go all the way to S 10,000.
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I just do a bunch of these.
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And what it's going to look like
over time is each of these--
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I'm going to make it
a dot, because I'm
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going to have to zoom out.
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So if I look at it like
this, over time-- it still
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has all the values that it
might be able to take on,
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2.75 might be here.
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So this first dot is
going to be-- this one
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right here is going
to be right there.
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And that second one is
going to be right there.
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Then that one at 3.5 is
going to look right there.
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But I'm going to
do it 10,000 times.
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Because I'm going
to have 10,000 dots.
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And let's say as I do it, I'm
going just keep plotting them.
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I'm just going to keep
plotting the frequencies.
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I'm just going to
keep plotting them
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over and over and over again.
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And what you're going
to see is, as I take
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many, many samples
of size 4, I'm
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going to have
something that's going
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to start kind of approximating
a normal distribution.
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So each of these dots represent
an incidence of a sample mean.
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So as I keep adding on
this column right here,
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that means I kept getting
the sample mean 2.75.
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So over time.
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I'm going to have
something that's
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starting to approximate
a normal distribution.
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And that is a neat thing about
the central limit theorem.
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So an orange, that's the
case for n is equal to 4.
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This was a sample size of 4.
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Now, if I did the same thing
with a sample size of maybe
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20-- so in this case, instead
of just taking 4 samples
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from my original crazy
distribution, every sample
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I take 20 instances
of my random variable,
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and I average those 20.
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And then I plot the
sample mean on here.
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So in that case,
I'm going to have
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a distribution that
looks like this.
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And we'll discuss
this in more videos.
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But it turns out if I were
to plot 10,000 of the sample
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means here, I'm going
to have something
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that, two things-- it's going
to even more closely approximate
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a normal distribution.
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And we're going to
see in future videos,
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it's actually going to
have a smaller-- well,
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let me be clear.
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It's going to have
the same mean.
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So that's the mean.
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This is going to
have the same mean.
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So it's going to have a
smaller standard deviation.
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Well, I should plot
these from the bottom
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because you kind of stack it.
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One you get one, then another
instance and another instance.
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But this is going to
more and more approach
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a normal distribution.
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So this is what's super
cool about the central limit
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theorem.
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As your sample size
becomes larger--
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or you could even say as
it approaches infinity.
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But you really don't
have to get that close
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to infinity to really get
close to a normal distribution.
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Even if you have a
sample size of 10 or 20,
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you're already getting very
close to a normal distribution,
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in fact about as
good an approximation
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as we see in our everyday life.
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But what's cool is we can start
with some crazy distribution.
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This has nothing to do
with a normal distribution.
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This was n equals 4, but if
we have a sample size of n
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equals 10 or n
equals 100, and we
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were to take 100 of these,
instead of four here,
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and average them and
then plot that average,
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the frequency of it, then we
take 100 again, average them,
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take the mean, plot
that again, and if we
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do that a bunch
of times, in fact,
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if we were to do that
an infinite time,
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we would find that
we, especially
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if we had an
infinite sample size,
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we would find a perfect
normal distribution.
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That's the crazy thing.
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And it doesn't apply just
to taking the sample mean.
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Here we took the
sample mean every time.
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But you could have also
taken the sample sum.
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The central limit theorem
would have still applied.
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But that's what's so
super useful about it.
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Because in life, there's all
sorts of processes out there,
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proteins bumping into
each other, people doing
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crazy things, humans
interacting in weird ways.
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And you don't know the
probability distribution
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functions for any
of those things.
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But what the central
limit theorem
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tells us is if we add a
bunch of those actions
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together, assuming that they
all have the same distribution,
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or if we were to take the
mean of all of those actions
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together, and if we were to plot
the frequency of those means,
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we do get a normal distribution.
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And that's frankly why the
normal distribution shows up
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so much in statistics
and why, frankly, it's
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a very good
approximation for the sum
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or the means of a
lot of processes.
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Normal distribution.
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What I'm going to show you in
the next video is I'm actually
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going to show you that this is
a reality, that as you increase
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your sample size, as
you increase your n,
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and as you take a
lot of sample means,
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you're going to have a frequency
plot that looks very, very
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close to a normal distribution.
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