Editing Central limit theorem (section)

{{Short description|Fundamental theorem in probability theory and statistics}}
{{use dmy dates|cs1-dates=ly|date=July 2023}}
{{Infobox mathematical statement
| name = Central Limit Theorem
| image = [[File:IllustrationCentralTheorem.png|300px|class=skin-invert-image]]
| field = [[Probability theory]]
| type = [[Theorem]]
| statement = The scaled sum of a sequence of [[Independent and identically distributed random variables|i.i.d. random variables]] with finite positive [[variance]] converges in distribution to the [[normal distribution]].
| generalizations = [[Lindeberg's condition | Lindeberg's CLT]]
}}

In [[probability theory]], the '''central limit theorem''' ('''CLT''') states that, under appropriate conditions, the [[Probability distribution|distribution]] of a normalized version of the sample mean converges to a [[Normal distribution#Standard normal distribution|standard normal distribution]]. This holds even if the original variables themselves are not [[Normal distribution|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 form it was only precisely stated as late as 1920.{{sfnp|Fischer|2011|p={{page needed|date=July 2023}}}}

In [[statistics]], the CLT can be stated as: let <math>X_1, X_2, \dots, X_n</math> denote a [[Sampling (statistics)|statistical sample]] of size <math>n</math> from a population with [[expected value]] (average) <math>\mu</math> and finite positive [[variance]] <math>\sigma^2</math>, and let <math>\bar{X}_{n}</math> denote the sample mean (which is itself a [[random variable]]). Then the [[Convergence of random variables#Convergence in distribution|limit as <math>n\to\infty</math> of the distribution]] of <math>(\bar{X}_n-\mu) \sqrt{n} </math> is a normal distribution with mean <math> 0 </math> and variance <math>\sigma^2</math>.<ref>{{Cite book |last1=Montgomery |first1=Douglas C. |title=Applied Statistics and Probability for Engineers |edition=6th |last2=Runger |first2=George C. |publisher=Wiley |year=2014 |isbn=9781118539712 |page=241}}</ref>

In other words, suppose that a large sample of [[Random variate|observations]] is obtained, each observation being randomly produced in a way that does not depend on the values of the other observations, and 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 is 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]].