Central Limit Theorem

The central limit theorem is one of the most important results in statistics.

Roughly speaking, it says that the sample mean has an approximately normal distribution when the sample size is large – even when the underlying population itself is not normally distributed.

This surprising fact is enormously useful. We can now use the familiar normal distribution to make probability calculations in a huge range of situations, and it will later underpin many of our methods for estimation, confidence intervals, and hypothesis testing.

We first develop the intuition and formal statement of the theorem, before looking at some empirical results to see just how quickly the normal approximation can emerge in practice.