The central limit theorem: why averages make bells

Heights, measurement errors, exam averages and polling results all tend to follow the same bell-shaped curve, even though the things being added up look nothing like a bell. A single die roll is flat; the average of many dice is a bell.

The central limit theorem says this is no coincidence: average enough independent samples of almost anything and the averages pile up into a normal distribution, with a spread you can predict exactly.

population

Where the samples come from, with mean $\mu$ and standard deviation $\sigma$. It can have any shape.

sample of size n

$n$ independent draws from the population.

sample mean x̄

The average of one sample. Draw many samples and you get many means.

standard error

How spread out the means are: $\sigma / \sqrt{n}$.

The population is lopsided, but the averages are not: big and small draws partly cancel, so averages land near $\mu$ more often than far from it, and they fall away symmetrically on both sides.

Use the slider to change $n$ and the buttons to change the population.

• n = 1: each "average" is a single draw, so the histogram just copies the population. The theorem needs averaging.
• n = 2 to 5 on the skewed population: watch the lopsidedness fade.
• Coin flips: the averages can only be $k/n$, so the histogram is spiky, but the spikes still trace a bell.

• The means of samples of size $n$ cluster around $\mu$ with spread $\sigma/\sqrt{n}$.
• For moderate $n$ their distribution is close to normal, whatever the population looks like.
• This is why confidence intervals, A/B tests and polls can use the normal curve, and why averaging more data helps only like $\sqrt{n}$.