Central Limit Theorem

The central limit theorem is a foundational result in probability and statistics stating that the sum or average of a large number of independent, identically distributed random variables tends toward a normal distribution, regardless of the shape of the underlying distribution being sampled. Formal versions of the theorem were proved across the eighteenth through early twentieth centuries by mathematicians including Abraham de Moivre and Pierre-Simon Laplace, with Aleksandr Lyapunov giving a rigorous modern proof in 1901 using characteristic functions. The theorem is the reason the normal distribution appears so often in statistical practice and underlies most standard methods of statistical inference, including confidence intervals and hypothesis tests.