The most important result about sample means is the Central Limit Theorem. Simply stated, this theorem says that for a large enough sample size \(n\), the distribution of the sample mean \(\bar{x}\) will approach a normal distribution. This is true for a sample of independent random variables from any population distribution, as long as the population has a finite standard deviation \(\delta\).
A formal statement of the Central Limit Theorem is the following:
If \(\bar{x}\) is the mean of a random sample \(X_1, X_2, \dots , X_n\) of size \(n\) from a distribution with a finite mean \(\mu\) and a finite positive variance \(\delta^2\), then the distribution of \(W = \frac{\bar{x} - \mu}{\frac{\delta}{\sqrt{n}}}\) is \(\mathcal{N}(0,1)\) in the limit as \(n\) approaches infinity.
This means that the variable \(\bar{x}\) is distributed \(\mathcal{N}(\mu, \frac{\delta}{\sqrt{n}})\).
