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