(At a regional public university)
This is from a “teacher evalaution” (of me) by a tenured professor who regularly taught statistics.
First, here is a little background about the class. (Math now, then I will try to explain non-mathematically.) I was talking to the class about the sample mean. The previous day, I had showed them that the sample mean of iid normally distributed variables is normal with the same mean and the smple mean’s sigma equal to sigma/sqrt(n). (A trivial computation) The tenured observer wrote that he thought I was talking about,
“…the Central Limit Theorem: that is, the sample mean’s are normally distributed with the same mean and with the samples mean’s sigma equal to sigma/sqrt(n)…”
For a simple explanation of the Central Limit Theorem, see “Applications and Examples” in the wikipedia article .
After that, try this, to convince yourself that the CLT makes sense.
Imagine an experiment where all the results are numbers between 0 and 1, and are equally likely (uniformly distributed). Now, imagine doing the experiment twice in a row and taking the average. The average is between 0 and 1 again, but are all the numbers now equally likely? To really see what numbers are most likely for the averages, do the experiment a hundred times and plot the averages. Are all the numbers between 0 and 1 now equally likely? Do you think that the probability of getting an average of zer0 is the same as getting an average of .5? (To simplify, further, suppose 0, .5, and 1 are the only possiblities, and they are equally likely. In repeating the experiment a 100 times, how many ways are there of getting an average of 0 or 1, or even very close? of something close to .5?)
If you graph the likelihood of the numbers for any n, you will see that as n gets bigger, the curve will look more and more like a bell curve (but never exactly), as n gets bigger. That, in a nutshell, is the idea of the Central Limit Theorem. What is critical here? That the initial probabilities do not have to be normal and that the sample means approach annormal. That is a powerful result that is extremely useful in probability and statistics. Compare that with what the professor above thought the Central Limit Theorem was – and ask yourself if you think his students are going to have a handicap if they ever teach high school statistics? and more.