Central Limit Theorem Explorer

This works because averaging is essentially repeated convolution, and convolving a distribution with itself enough times smooths out its individual quirks — skewness and multi-modal humps decay faster than the spread does, so what's left increasingly looks Gaussian regardless of where you started. The only real requirement is a finite variance; a uniform, an exponential, and a two-humped bimodal source all get pulled toward the same bell curve once you average enough draws, just at different speeds — which is why n=30 became statistics' informal rule of thumb rather than an exact threshold.

As you keep drawing, notice the histogram of means gets narrower, not just smoother. Its spread is the standard error σ/√n, so precision improves only with the square root of the sample size — to halve the width you must quadruple n. That is the practical heart of the theorem: bigger samples pin down an average more tightly, but with steadily diminishing returns.

The theorem quietly assumes a finite variance, and one famous outlaw violates it: the Cauchy distribution, the ratio of two standard normals. Average a million Cauchy draws and the result follows the exact same Cauchy curve as a single draw — the bell never forms and extra data buys nothing. Its tails are so heavy that one monstrous value can dominate any average, however large.