# Chapter 3 The Central Limit Theorem

We now turn our attention to one of the most fundamental results in statistics: The remarkable Central Limit Theorem.

**The Central Limit Theorem (CLT)**

Let \(X_1, \ldots, X_n\) be a random sample from a distribution with finite mean \(\mu\) and finite variance \(\sigma^2\). For \(\overline{X}\) denoting the sample mean, if \(n\) is sufficiently large then \[\overline{X}\stackrel{\tiny \text{approx.}}\sim N\left(\mu,\frac{\sigma^2}{n}\right)\] where \(\stackrel{\tiny \text{approx.}}\sim\) denotes 'approximately distributed as'.

Normally, a sample size of approximately \(n = 30\) is considered to be 'sufficiently large'.

To help us understand this theorem, let's consider some simulated examples.