The Central Limit Theorem, by hand

The engine under the whole second half of the course: no matter how lopsided the population, the average of a sample piles up into a tidy bell curve centred on the true mean, with spread \(\sigma/\sqrt{n}\). Pick a population, drag the sample size \(n\), and watch it happen.

Try it: start at n = 1 (the sampling distribution is just the population — as skewed as ever), then climb. By n = 30 even the most lopsided population averages into a near-perfect bell, and the spread shrinks by the \(\sqrt{n}\) law.