Making the Bell Curve Rigorous

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

By 1900, the central limit theorem had two famous names attached to it and neither claim was actually complete. De Moivre had proved it for a single narrow case, repeated fair coin flips, in 1733. Laplace had generalized it in 1810, but by later standards not rigorously: his argument assumed conditions were well behaved without stating precisely what "well behaved" would have to mean in general, or proving that the conclusion held whenever those conditions failed to be met exactly. It took Aleksandr Lyapunov, a Russian mathematician better known in his own century for founding the theory of dynamical stability, to close the gap in 1901: he stated a precise, checkable condition, now called the Lyapunov condition, under which the sum of many independent random quantities converges to a normal distribution regardless of what the individual quantities' own distributions look like, and he proved it held. That generality is exactly why the theorem matters as much as it does across so much of applied statistics: it explains why so many unrelated kinds of noisy data, from measurement error to survey samples, tend toward the same bell-shaped curve, and it was Lyapunov who first said precisely when that tendency was guaranteed rather than merely observed.