Repeated samples from a non-normal population produce Gaussian sample means.
From the chapter: Chapter 5: Statistics
Glossary: central limit theorem, sampling distribution, standard error
Transcript
A sampling distribution describes how a statistic, such as the sample mean, varies across repeated samples from the same population.
The starting population is anything but normal. This bimodal distribution has two peaks and looks nothing like a bell curve.
Many samples of size two are drawn. Each one produces a single mean, plotted as a red dot. The histogram already starts to round at the edges.
The sample size jumps to ten. The histogram tightens, and a bell shape begins to emerge despite the bimodal source.
At thirty samples per draw, the histogram closely matches a Gaussian. The dashed curve is the theoretical limit predicted by the central limit theorem.
The standard error shrinks as one over the square root of n. Halving the spread requires four times the sample size.
That is the central limit theorem. Averages become Gaussian even when the underlying data is not. It is the engine behind confidence intervals and hypothesis tests.