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# Central limit theorem

AP.STATS:
UNC‑3 (EU)
,
UNC‑3.H.2 (EK)
,
UNC‑3.H.3 (EK)

## Video transcript

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 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 could be a continuous distribution or 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 can take on values 1 through 6 1 2 3 4 5 6 it's some kind of crazy dice that's very likely to get a 1 let's say it's impossible let me make that a straight line your very high likelihood of getting one let's say it's impossible to get a 2 let's say it's an ok likely to being a 3 or 4 let's say it's impossible to get a 5 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 we would look it'd 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 so let's say 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 size is 4 let's say I get a 1 let's say I get another one and let's say I get a 3 and I get a 6 so that right there is my first sample of sample size four and I know the terminology can get confusing because this is a sample that's made up of four samples but in 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 could easily confuse 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 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 what 2 point seven five two point seven five 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 three for a let's say 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 twos or fives over here so in this for the second sample of sample size for my sample mean for my second so my second sample mean is going to be 3 plus 4 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 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 so I'm going to literally take 4 samples so my sample is made up of 4 samples from this an 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 14 divided by 4 is what 3 a half three and a half and as I find each of these sample means so I eat for each of my samples of sample size for 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've plot this all on a frequency distribution so I say okay on my first sample my first sample mean was two point seven five so I'm plotting the actual frequency of the sample means I get for each sample so two point seven five I got at one time so I'll put a little plot there so that's from that one right there and I got the next time I also got a two point seven five that's a two point seven five there so I'll so I got twice so I'll plot the frequency right there then I got a three and a half so all the possible values I could have a three I could have a three point two five I could have a three and a half so then I have the three and a half 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 you know maybe I'll take ten thousand of them so I'm going to keep taking these samples so I go all the way to s you know ten thousand I just do a bunch of these and what's going to look like over time is each of these I'm going to make it a dot because I'm gonna have to zoom out so if I look at it like this over time it's tell us all the values that it might be able to take on you know two point seven five might be here so this first dot 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 that one at three point five is going to look right there but I'm gonna do it ten thousand times going to have ten thousand dots 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 four 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 two point seven five so over time I'm going to have something that's starting to approximate a normal distribution and that is the neat thing about the central limit theorem so the central limit to end this was the case this was the case for so in orange that the case for n is equal to four this was a four sample size of four now if I did the same thing with the sample size of maybe 20 so in this case instead of just taking four samples from my original crazy distribution every sample I take twenty instances of my random variable and I average those twenty 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 the sample 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 distribution and we're going to see in future videos it's actually going to have a smaller well let me let me be clear it's going to have a small it's going to have the same mean so that's the mean this is going to have the same mean that's going to have a smaller standard deviation so I want to well I should plot these from the bottom because you kind of stack it when you get one and another instance in another instance but this is going to more and more approach a normal distribution so the the reality is and this is what's super cool about the central limit theorem as your sample size as your sample size as your sample size becomes larger 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 you know 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 isn't this has nothing to do with the normal distribution but if we have a sample size this was N equals 4 but if you have a sample size of N equals 10 or N equals 100 and we were to take a hundred of these instead of 4 here and average them and then plot that average the frequency of it then we do take 100 again average them take the mean plot that again and if we would do that a bunch of times in fact if we would do that an infinite time we would find that we 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 cent 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 process is out there proteins bumping into each other you know people doing crazy things humans interacting in weird ways and you don't know the probability distribution functions for any of those things but the central limit theorem tells us is that 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 distribution and that's frankly why the normal distribution shows up so much 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 showing 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