Why does the bell curve keep showing up, even when nothing is bell-shaped?

Drop a bead into a forest of pegs and, at each peg, it bounces left or right. Do that with hundreds of beads and they collect in bins at the bottom, the columns forming a bell curve. This is the Galton board, a device invented by Francis Galton to demonstrate the central limit theorem: with a large enough sample, the binomial distribution, which counts left and right bounces, approximates a normal distribution. Lay Pascal's triangle over the pegs and it shows the number of different paths a bead can take to each bin.

The theorem itself says something remarkable. Take the average of many independent observations. Repeat that many times, and the collection of averages will closely approximate a normal distribution, even if the original variables are not normally distributed. Roll many dice, and the sum or average of the numbers is well approximated by a normal distribution. Since real-world quantities are often the balanced sum of many unobserved random events, the theorem partly explains why the normal distribution is so common.

Its history is a story of being ignored. Abraham de Moivre postulated the first version in 1733, using the normal distribution to approximate the number of heads in many tosses of a fair coin. The finding was far ahead of its time and nearly forgotten until Laplace rescued it from obscurity in 1812. Only at the end of the nineteenth century, in 1901, did Aleksandr Lyapunov define it in general terms and prove precisely how it worked.

Even Alan Turing met the theorem late. His 1934 Fellowship Dissertation at King's College, Cambridge, proved a result similar to Lindeberg's central limit theorem, and only after submitting it did he learn it had already been proved.