The Central Limit Theorem

Summary :

This chapter presents the Central Limit Theorem as a rigorously demonstrated theorem rather than a theory, comparable in certainty to the Pythagorean theorem. It explains why means are so central to statistics, how the theorem describes the distribution of sample means regardless of the original population's shape, and what sample size counts as large enough.

Why the theorem matters

The chapter argues that means are central to statistics because they provide both a middle ground for comparison and are easy to calculate, and it stresses that the Central Limit Theorem is a genuine theorem, mathematically demonstrated with the same rigor as the Pythagorean theorem, not merely a proposed way of looking at the world.

Sample means and the normal distribution

The theorem concerns drawing finite samples of size n from a population with a known mean and known standard deviation; if n is large enough, the distribution of the resulting sample means will tend to follow an approximately normal distribution regardless of the shape of the original population's distribution, a result the chapter calls astounding precisely because it does not require knowing that original shape.

How large a sample is large enough

The chapter addresses the practical question of what sample size counts as large enough for the theorem to apply well, generally citing a sample size of at least 30 unless the underlying population is already known to be normal, and noting that a population further from normal requires a larger sample before the sample-mean distribution becomes reliably normal.