Central limit theorem

The central limit theorem says that if you take samples of size n from almost any population and average each sample, the distribution of those averages approaches a normal distribution as n grows — regardless of the population's shape.

mean of the sample means → μ standard error → σ / √n shape → normal, as n grows

Try the coin flip

Set the population to the coin flip: individual values are only ever 0 or 10, about as far from a bell curve as a distribution can be. With n = 1 the means histogram shows the same two spikes. Raise n to 5 and a shape appears. At n = 30 it is convincingly normal. Nothing about the coin changed — only how many of them you average.

Why √n

Averaging cancels errors, but not linearly. Independent variances add, and standard deviation is the square root of variance, so the spread of the mean falls as √n. This is the reason polling precision is expensive: halving the margin of error requires quadrupling the sample. Going from 1,000 to 2,000 respondents only improves precision by about 29%.

Why this matters

It is the reason nearly all of classical statistics works. Confidence intervals, t-tests and regression standard errors all rely on the sampling distribution of an estimate being approximately normal — and the CLT delivers that even when the underlying data is nothing of the sort. You never needed the population to be normal; you needed the estimate to be.

When it fails

The theorem requires finite variance and independent draws. Heavy-tailed distributions such as the Cauchy have no finite variance, and their sample means never settle down — the average of a million draws is no more reliable than a single one. Dependence between observations (time series, clustered survey data) breaks it too, which is why those need their own machinery.

The common “n ≥ 30 is enough” rule of thumb is a rough guide, not a theorem. Strongly skewed populations need considerably more; the exponential population above is a good place to see that for yourself.