Distribution of Sample Mean and Variance

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be independent random variables from a distribution having mean \(\mu\) and variance \(\sigma^{2}\). From the central limit theorem,

\[ \begin{aligned} \frac{X_{1} + X_{2} + \cdots + X_{n} - n\mu}{\sigma \sqrt{n}} \sim \mathcal{N}(0, 1) \end{aligned} \]

or, the sum of the random variables follows the distribution of a standard normal as the value of \(n\) becomes large. Typically, the property starts to manifest as soon as \(n\) becomes around 30.