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I get confused while reading the statement of Central Limit Theorem.

CLT states that -If we are sampling from a population that has an unknown probability distribution, the sampling distribution of the sample mean will still be approximately normal with mean u and variance sigma^2/n , if n is large.

What does the n represents here? Is it a sample size or the number of samples. Suppose I have 10 samples of 40 observations each. In this case my n will be 10 or 400(10*40)?

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  • $\begingroup$ $n$ is the number of observations for your sample. $\endgroup$ Nov 16, 2022 at 11:03
  • $\begingroup$ @Shivam Bansal Then you have 10-many, n=40 samples. $\endgroup$ Nov 16, 2022 at 11:35

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The $n$ you are mentioning is the number of random samples. Normally you work with independent and identically (i.i.d) distributed random variables (which is a fancy way to say "there is no significant difference on how you sample $X_i$ and $X_j$", so you don't skew on one way or another. )

So the $n$ in your question is basically "how many $X_i$" you take. You are interested in knowing "something" of your population. In textbook, the classic example is the expected value (average). But your population is either infinite, or you don't know all the elements, so you cannot calculate it explicitly.

Here is where the CTL comes into the game. You cannot calculate that "something" from your whole population, but you can calculate that from a randomly selected sample.

Example: you cannot calculate the average height of all the male humans in the world, because it is unfeasible for you to measure every living male right now. But you can take random samples of let's say 100 people, and you can calculate their average height. Informally, the CTL states that, the more samples you take (and calculate their average height of each sample and then the average of the averages), that final value will slowly but surely (by the law of large numbers) get closer to the real value of the average height of all the living males.

For your value to get more accurate, you need to take a big amount of samples. Now, you want to make some statistical analysis of the average of averages (like saying "We are at least 95% confident that this value is at most $X$ units away from the real average"). You need to have a distribution to work with (specially if you want to do hypothesis tests). Here is where the CTL tells you "hey, do this long enough, and you can use the normal distribution".

Edit: I forgot to mention that the CTL doesn't work with all the statistics. Average (mean) is the shown in textbook, and some others you can transform them to the average. But there are a lot of them for which it doesn't work (example: standard deviation, which is why you divide by $n-1$ sometimes and by $n$ other times).

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  • $\begingroup$ Care is needed for comments about confidence interval. Any true parameter lies in a CI with either probability $1$ or $0$. %95-CI means If you come with m-many CI derived from samples-of-n, then %95 of m-confidence intervals will contain the true parameter, as m grows. $\endgroup$ Nov 16, 2022 at 11:30

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