Central Limit Theorem

The central limit theorem explains why the normal distribution appears so often in statistics. Even if individual observations follow an unusual distribution, the averages of large samples of them will be approximately normally distributed, as long as certain conditions are met.

For example, the outcome of rolling a single die is evenly spread from 1 to 6, which is not a bell shape at all. But if you roll 30 dice many times and record the average each time, those averages will form a pattern very close to a bell-shaped normal distribution, centred on 3.5.

The theorem is the basis for many statistical methods, such as confidence intervals and hypothesis tests, which rely on sample averages being approximately normal. It allows analysts to draw conclusions about a whole population from a sample.

In finance, it is sometimes used to justify modelling returns over longer periods as roughly normal, since they are made up of many smaller daily returns. However, the theorem relies on assumptions, such as independent observations without extremely fat tails, which do not always hold in financial markets.