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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 can 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 could 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 discrete 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 one.

• Let's say it's impossible-- well,

• let me make that a straight line.

• You have 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 to 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 the 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 would

• look-- it would be that far and that

• far above and below the mean.

• But that's my discrete 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.

• 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 four samples from this.

• So let's say the first time I take four samples--

• so my sample sizes is four-- let's say I get a 1.

• Let's say I get another 1.

• And 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 the sample that's made up of four samples.

• But then 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

• And the sample size tells you how many you actually

• But the terminology can be very confusing,

• because you could easily view one of these as a sample.

• But we're taking four samples from here.

• We have a sample size of four.

• And what I'm going to do is I'm going to average them.

• So let's say the mean-- I want 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.

• It's impossible for this distribution.

• The chance of getting a 2 or 5 is 0.

• So I can't have any 2s or 5s over here.

• So for the second sample of sample size 4,

• 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--

• so I'm going to literally take 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, and 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 and 1/2.

• 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 and 1/2.

• So all the possible values, I could have a three,

• I could have a 3.25, I could have a 3 and 1/2.

• So then I have the 3 and 1/2, 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 it 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,

• 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.

• Then that one at 3.5 is going to look right there.

• But I'm going to do it 10,000 times.

• Because I'm going to have 10,000 dots.

• And let's say as I do it, I'm going 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 an orange, that's the case for n is equal to 4.

• This was a 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