convergence

central limit theorem in action

The Central Limit Theorem: the mean of n independent samples approaches a normal distribution as n grows — regardless of the source distribution's shape.

σ_means = σ_source / √n

This is why confidence intervals and hypothesis tests work. The spread of sample means shrinks predictably as 1/√n, making estimation more precise with larger samples.

live simulation

source distribution

sample means · n = 1

mean of means
std of means
theoretical std
samples 0