Central Limit Theorem

The central limit theorem says that if you average a large enough number of independent observations, the distribution of that average is approximately a normal bell curve — no matter what shape the original data had. Skewed, lumpy, or weird raw data still produces nicely normal averages.

This is arguably the most important result in statistics, because it is why so many methods (confidence intervals, many hypothesis tests) can assume normality even when the underlying data is not normal. Two cautions: it is about the average (or sum), not the individual data points, and “large enough” can mean a lot of observations when the data is very skewed.