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.