What does the central limit theorem say?

1lopsided data

Here is a lopsided population: minutes people wait for a bus. Many short waits, a few very long ones. The average wait is four minutes.

2one average

Now grab thirty people at random and average their waits. One sample like that gave three point nine minutes. Mark that average on a new line.

3averages pile up

Do it again and again. Each new sample gives a different average, but they crowd around four minutes, with few far out. A pile like that is a distribution of its own.

4the pile's shape

The central limit theorem says what that pile looks like. Take samples of size n from any population with a mean and a finite spread. The averages form a bell.

5n = 5

With samples of five, the pile is already narrower and its hump has moved in from zero toward four. It is still lopsided, though, leaning to the right.

6n = 30

With samples of thirty, the lopsidedness is nearly gone: a tall, narrow, nearly symmetric bell around four minutes, the true average. The original data never changed.

7spread shrinks

And the spread shrinks like one over the square root of n. Wait times spread four minutes, so thirty-person averages spread only four over root thirty: about seven tenths of a minute.

8the whole idea

So the whole theorem in one picture: any lopsided population goes in, you average n values, and the averages come out bell-shaped around the true mean.

9fine print

The fine print: samples must be independent, the population needs a finite spread, and thirty is a rule of thumb. Very skewed data wants more.