Let $X_1, X_2, X_3 \ldots $ be independent identically distributed random variables where $\mathrm{E}[X_i] = \mu$, $\mathrm{Var}[X_i] = \sigma^2$ for all $i$ then the average \[\bar{X} = \frac{1}{n}\sum\limits_{1}^{n}X_i\] of the first $n$ random variables are approximately normally distributed with mean $\mu$ and variance $\dfrac{\sigma^2}{n}$ in the following manner: \[\lim_{n\rightarrow \infty }P\left[ \frac{\bar{X}_n - \mu}{\sqrt{\frac{\sigma^2}{n}}} \leq z \right] = F(z).\]