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what does the central limit theorem say
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.
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.
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.
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.
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.
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.
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.
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.
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.
Averages forget the shape of what they average: add enough of anything and a bell appears.