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

Let X be a random variable, and let X_1, X_2, \ldots , X_n be a random sample for X, such that each X_i has a distribution identical to that of X itself. Let \overline{X} be the sample mean; in other words, let \overline{X} be equal to \frac{\sum X_i}{n}. Because each X_i is a random variable, \overline{X} is also a random variable. The Central Limit Theorem observes several important facts about the distribution of \overline{X}:

  1. The distribution of \overline{X} is approximately normal, even when the underlying distribution X is not.
  2. The expected value of the \overline{X} is equal to the expected value of X.
  3. As the sample size n increases, the variance of \overline{X} approaches zero.

See also