Central limit theorem

In probability theory, the central limit theorem (CLT) states that, under appropriate conditions, the distribution of a normalized version of the sample mean converges to a standard normal distribution. This holds even if the original variables themselves are not normally distributed. There are several versions of the CLT, each applying in the context of different conditions.

The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920,^[1] thereby serving as a bridge between classical and modern probability theory.

An elementary form of the theorem states the following. Let ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ denote a random sample of ${\displaystyle n}$ independent observations from a population with overall expected value (average) ${\displaystyle \mu }$ and finite variance ${\displaystyle \sigma ^{2}}$, and let ${\displaystyle {\bar {X}}_{n}}$ denote the sample mean of that sample (which is itself a random variable). Then the limit as ${\displaystyle n\to \infty }$ of the distribution of ${\displaystyle ({\bar {X}}_{n}-\mu )/\sigma _{{\bar {X}}_{n}},}$ where ${\displaystyle \sigma _{{\bar {X}}_{n}}=\sigma /{\sqrt {n}},}$ is the standard normal distribution.^[2]

In other words, suppose that a large sample of observations is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and that the average (arithmetic mean) of the observed values is computed. If this procedure is performed many times, resulting in a collection of observed averages, the central limit theorem says that if the sample size was large enough, the probability distribution of these averages will closely approximate a normal distribution.

The central limit theorem has several variants. In its common form, the random variables must be independent and identically distributed (i.i.d.). This requirement can be weakened; convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations if they comply with certain conditions.

The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.