Let $X_1, X_2, X_3, \dots, X_n$ be a random sample with mean $\mu$ and variance $\sigma^2$. If $n$ is sufficiently large, $\bar{X} = \frac1n \sum_i^n X_i$ has approximately a normal distribution $\bar{X} \approx N(\mu, \frac{\sigma^2}{n})$ The larger the value of $n$ the better the approximation. A typical rule of thumb is $n \ge 30$, but it does depend on the underlying distribution. A more symmetrical distribution tends to require fewer samples than a more skewed distribution. Technically, the distribution of $\bar X_n$ is [[asymptotically normal]], or [[converges in distribution]] to the normal distribution as $n$ goes to infinity. The central limit theorem applies regardless of the underlying distribution, whether continuous or discrete. For example, let $\bar X$ be the sample mean for a random sample of size 100 from the [[gamma distribution]] $\Gamma(3,2)$. The mean is the expected value for any one of the random variables $E[\bar X] = E[X_1] = \frac{\alpha}{\beta} = \frac32$ and the variance is the variance for any one of the random variables over $n$ $V(\bar X) = \frac{Var(X_1)}{n} = \frac{\alpha / \beta}{n} = \frac{3/4}{100}$ By the CLT, $\bar X$ is normally distributed $N \sim (\frac34, \frac{3}{400})$