The central limit theorem

Exp(1): right-skewed. Mean 1, sd 1. Finite variance — the theorem applies.

means drawn

0

last sample mean

—

sd of the means

—

theory σ/√n

0.354

Each draw averages n independent samples and drops one mean into the lower histogram. Both plots share the axis; the tick at 1 is the population mean. With finite variance the pile rounds off and its sd tracks σ/√n. Gaussianity of the data is never required — finite variance is.