Central Limit Theorem

Take many samples of size n from almost any distribution and compute each sample's mean. The Central Limit Theorem says those means form an approximately normal distribution centred on the true mean, with spread $\sigma/\sqrt{n}$, the standard error.

That $\sqrt{n}$ is why precision is expensive: to halve your uncertainty you need four times the data.

It is also why most statistical tests work at all: we can reason about the sampling distribution of a mean even when the raw data is skewed. Watch the histogram of means turn into a bell curve in the simulation, even from a lopsided source.