Central Limit Theorem

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

Measure the heights of enough adults, the errors in enough repeated measurements of the same quantity, or the outcomes of enough coin flips added together, and the same bell-shaped curve keeps appearing, however different the underlying process looks up close. The central limit theorem is the reason: whenever an observed quantity is really the sum, or average, of a large number of small, independent random contributions, the shape of any one contribution barely matters, and the sum settles toward a normal distribution regardless. The theorem did not arrive all at once. In 1733, Abraham de Moivre, a French Huguenot mathematician who had fled religious persecution for London and made his living partly by calculating odds for gamblers, found that the binomial distribution, the distribution of the number of heads in many coin flips, could be closely approximated by what is now called the normal curve when the number of flips is large. He published the result as a way of computing binomial probabilities more easily, not as a general law about randomness. It was Pierre-Simon Laplace, in 1810, who took de Moivre's specific approximation and generalized it into something closer to the modern theorem, showing that the same normal-curve limiting behavior held for sums of many kinds of independent random quantities, not only coin flips. Even Laplace's version fell short of full rigor by later standards: it was the Russian mathematician Aleksandr Lyapunov who, in 1901, finally pinned down precise, general conditions under which the theorem is guaranteed to hold, closing a gap that had stood for over a century. The result these three built in stages is now one of the load-bearing facts of applied statistics: it is the reason a sample average becomes more reliable, and more nearly normal, the larger the sample, almost regardless of what is being measured.

This article records tradition as it has been passed down and reported. Its sources are not yet part of the atlas's verified catalogue.

A theorem can be true, useful, and widely believed for a hundred years before anyone states precisely when it actually applies. That is what happened to the central limit theorem between 1733 and 1901. Abraham de Moivre's original 1733 result covered exactly one case: the binomial distribution, the count of heads in a long run of coin flips, approximated by the normal curve as the number of flips grows. It worked, and it was useful for computing gambling odds, but it said nothing about any other kind of randomness. Pierre-Simon Laplace, working seven decades later, extended the claim far beyond coin flips, to sums of many kinds of independent random quantities, and effectively promoted it from a computational trick for one distribution into a general law about sums of randomness. But Laplace's own derivation relied on the mathematical standards of his day, which fell short of what a modern proof requires: exactly what conditions a sequence of random variables must satisfy for their sum to converge to a normal distribution was still not stated with full rigor. It took until 1901 for the Russian mathematician Aleksandr Lyapunov to supply conditions, now called the Lyapunov condition, precise enough to prove the theorem in general, opening the door to the many further variations and generalizations of the central limit theorem that followed across the twentieth century as probability theory itself became a rigorous, measure-theoretic subject. The theorem's own history is therefore a small case study in how mathematics actually develops: not usually as one person's single flash of insight, but as successive mathematicians taking a useful but locally proven claim and, often across generations, finding out how far it really extends and how firmly it can be nailed down.