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The Central Limit Theorem states that the the distribution of \bar{X} approaches normality as n increases, regardless of what the distribution of X is.
Stated more formally, for samples of size n drawn from a distribution with mean \mu and finite variance \sigma^2, the distribution of the sample mean is approximately N\left(\mu, \frac{\sigma^2}{n} \right) for sufficiently large n.
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Glossary

central limit theorem

The mean of a random sample of size n from a distribution of mean m and variance v will be normally distributed with mean m and variance v/n, for large n.

limit

the value that a function f(x) approaches as the variable x approaches a value such as 0 or infinity

mean

the sum of all the members of the list divided by the number of items in the list.

sample

A collection of sampling units drawn from a sampling frame.

variance

The square of the standard deviation; a measure of dispersion.

Full Glossary List