Central limit theorem

The central limit theorem states that if $X_1, X_2, X_3, \cdots, X_n$ is a random sample from any population with mean $\mu_X$ and variance $\sigma_X^2$, then the sample mean approaches $\bar{X} \sim N(\mu_X,\frac{\sigma_X^2}{n})$, thus is normal distributed around the mean of all X, but with lower variance as the sample size increases.