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.