Central Limit Theorem: What Exactly Does It Do?

On this site I am going to illustrate how the Central Limit Theorem works.

Let us remind ourselves what the theorem says: Suppose you draw a random sample of numbers from any distribution f. If you plot this sample as a histogram, then the histogram will resemble the distribution f. For example, if f is the uniform distribution, then your histogram will have the familiar rectangular shape. But what about the distribution of the sample average? Suppose we calculated the average of our sample, and wrote down the result. Then we draw another random sample, compute the average, and write it down. If we do this many times, then we can draw a histogram of the sample averages. It turns out that this histogram has the bell shape of the normal distribution - regardless what the underlying distribution f was. And that is what the Central Limit Theorem tells us: the distribution of a sample average approaches a normal distribution in the limit - the bigger the sample, the 'more normal' the sample average gets. Wikipedia knows more than me about the Central Limit Theorem (and everything else that exists) so follow this link to find out more.