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## AP®︎/College Statistics

### Course: AP®︎/College Statistics>Unit 9

Lesson 2: The central limit theorem

# Central limit theorem

Introduction to the central limit theorem and the sampling distribution of the mean. Created by Sal Khan.

## Want to join the conversation?

• If the sample-size approaches infinity (or the size of the population), wouldn't the form of the distribution of the sample means go towards a distribution with a deviance of 0, practically becoming useless? I hope this makes sense. • You are correct, the deviation go to 0 as the sample size increases, because you would get the same result each time (because you are sampling the entire population). However, the deviation of the sampled means is not an indicator of the deviation of the entire population (as opposed to the mean of the sampled means, which IS an indicator of the mean of the entire population).
• What is a typical sample size that would allow for usage of the central limit theorem • In practice, "n = 30" is usually what distinguishes a "large" sample from a "small" one. In other words, if your sample has a size of at least 30 you can say it is approximately Normal (and, hence, use the Normal distribution). If, on the other hand, your sample has a size less than 30, it's best to use the t-distribution instead.
• Can the said "crazy dice" really approach a normal distribution? I mean, its sample average can never go above six or below one. Such results would be possible in any normal distribution, which covers the whole spectrum from "minus infinite" to "plus infinite". • The sample average stays in [1,2,3,4,5,6] if the sample size is 1 (because we only roll the 'crazy die' once). But we're increasing the sample size (rolling the 'crazy die' more times) which will increase the range of values of the sample average. I think that this helps because it improves the 'resolution' of the distribution so the differences between it and a normal distribution become smaller and smaller as the sample size increases.
• I am slightly confused here... Any help will be very appreciated.

1. The normal distribution obtained after averaging a large number of samples - Is it a good representation of the original distribution at all?

For example, if we try to deduce the probability of getting a 4.5 to 5.5 from the resultant normal distribution, it will give us a finite value whereas the original distribution clearly indicates that the probability for this outcome is zero. What am I missing here?

And if the final normal distribution is NOT a good representation of the original distribution, what is it's purpose in that case?

2. At , Sal says that if we increase the sample size, the standard deviation will be even smaller. Does this continue indefinitely or does the value of standard deviation stabilize somewhere? I mean if our sample size approaches infinity, will the standard deviation start approaching zero? If yes, will this steep distribution still be useful?

Many Thanks! • 1. No. But you may have misunderstood something. We are not averaging a large number of samples, rather, we are obtaining the averages from many repeated samples. The distribution of the sample averages is the Normal distribution we obtained. It does not represent the original distribution well. But it's not supposed to do so! This Normal distribution is the distribution of the sample mean. Its use it to let us talk about the probability of the sample mean being in a given interval, better understanding the population mean, and so forth.

2. There is no lower bound. If we get an astronomically large sample size, the standard deviation will be astronomically small. The Normal distribution is still useful - it's getting narrower and narrower, which lets us be more and more precise when we use it to try and talk about the population mean.
• why does this happen? Why does most behavior/characteristics etc., when sampled and plotted, result in a normal distribution? • It is when the sample means from multiple samples are plotted that you get an approximately normal distribution. There are much more technical explanations, but what I tell my Intro to Stats students is that calculating a mean from any sample is going to help even out the high and low values in a sample. Even a mean from a very small sample of n = 5 or 10 has this effect, just by the nature of the mean calculation. It makes intuitive sense that calculating means will give you values that are closer to the overall mean and more tightly distributed than the original data. Eventually even for bimodal and skewed distributions, as n increases, you can see the distributions of the means move more and more toward being unimodal and symmetrical because the mean has the effect of pulling toward the center.
• When should you use sigma or 'mu'? Also, what's the difference between an average and mean? • Usually, sigma and mu are used for the standard deviation and the mean of a population, whereas S and X bar are used for the standard deviation and mean of a sample.

The word 'average' is a bit more ambiguous. Average can legitimately mean almost any measure of central tendency: mean, median, mode, typical value, etc. However, even "mean" admits some ambiguity, as there are different types of means. The one you are probably most familiar with it the arithmetic mean, although there is also a geometric mean and a harmonic mean.
• Can the central theorem be abstractly proved? Or one gets to it only by observing an infinitely large number of samples? • I have a question about the usefulness of the Central Limit Theorem. In this video, the normal distribution curve produced by the Central Limit Theorem is based on the probability distribution function. I assume that in a real-world situation, you would create a probability distribution function based on the data you have from a specific sample. Then you could use a computer program to create a curve based on hypothetical subsamples that follow this distribution, and you could use that curve to calculate p values and so on. But I assume that the way you have obtained the probability distribution function is by observing the results of a finite number of observations. For example, maybe the one in the video was obtained by rolling the crazy die 100 times. What if it were the case that if you actually rolled the crazy die 1,000 times, you would have gotten a a couple of 2's, but no 5's. Then your normal curve would actually be lower than you originally thought. This could make a difference in the significance values that you get from a t-test. • There are two things to keep in mind here:
1. The distribution of the original data
2. The (sampling) distribution of the sample mean.

Generally, we do not know #1. With the die example, we know it - the distribution will be Uniform with values 1-6, all equally weighted (unless the die was loaded). However, in generally, we don't necessarily know the distribution. And even if we knew the form of the distribution (say - Normal, or Exponential, or Poisson, etc.), we typically do not know the parameters of the distribution, so we still can't calculate probabilities and so forth.

So we collect a sample of data, and yes, you are correct, we can get a sense of the population distribution from this sample. But here we hit a snag: what if the data are such that the distribution look somewhat Normal, but not fully? Can we just use the Normal distribution to calculate probabilities? What if the data follow some other - similar - distribution, such as the Laplace distribution:

https://en.wikipedia.org/wiki/Laplace_distribution

It can sometimes be difficult to tell these distributions apart just based on a potentially small sample. We could just assume the data are "Normal enough" and move forward, but is that really appropriate? How will that impact the results? How much error will get introduced into our results because of this?

If we shift our focus to the sample mean, then the CLT can remove some of these doubts. Because the sampling distribution of the sample mean converges to the Normal distribution when the sample size gets large enough. So we don't even need to care about the distribution of the original data, we can just think about the distribution of the sample mean. Since the two distributions have the same population mean, µ, this means that we can get information about µ using the sampling distribution of the sample mean, instead of the distribution of the original data.
• Does this mean the average of the sample means is the same as average of the original population?
(1 vote) • This works for rolling dice, but let me ask a question where the class is not a number: specifically,

Suppose I have the result of 10000 chess games (only wins), where I have the frequency of winning when one of the eight pawns is moved (let's ignore knight moves). Let us only consider this for white

The pawns on a chess board are placed on columns marked by alphabets that run from a to h.

If the frequency of wins by pawn is:

a 300
b 500
c 1700
d 3000
e 3000
f 600
g 700
h 200

When I take a sample size of say 30, how do I average it out? In Sal's case, he had a number on which the dice landed (1-6). In my case, I don't have these numbers. So, how do I find the mean? 