The central limit theorem
Repeated sampling turns a mathematical theorem into something you can watch happen.
Build a sampling distribution
Each bar describes the frequency of a sample mean, not an individual observation. Generate 1,200 independent sample means and change the sample size.
Mean of sample means—
Observed SD of means—
Theoretical standard error—
The population need not be normal. Independent observations and a suitable finite-variance model are important assumptions. Simulations fluctuate from run to run.