# Central Limit Theorem

*1,010*pages on

this wiki

The **Central Limit Theorem** identifies the distribution of the sample mean and is arguably the most important theorem in probability theory.

Let be a random variable, and let be a random sample for , such that each has a distribution identical to that of itself. Let be the sample mean; in other words, let be equal to . Because each is a random variable, is also a random variable. The Central Limit Theorem observes several important facts about the distribution of :

- The distribution of is approximately normal, even when the underlying distribution
*is not*. - The expected value of the is equal to the expected value of .
- As the sample size increases, the variance of approaches zero.