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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.