Statistics · Means become Gaussian
The central limit theorem
Sample from a skewed population. The histogram of sample means tightens and rounds as n grows.
Population — Exp(1), skewed
Sampling distribution of the mean (n = 8)
Each draw takes n independent exponential samples and records their mean. The population is right-skewed. The means are not. That is the theorem: finite variance plus independence is enough. Gaussianity of the data is not required.
- means drawn
- 0
- last sample mean
- —
Theory: E[X] = 1, Var(X) = 1, so the mean has SE = 1/√n. Raise n and the lower histogram tightens around 1.