Central Limit Theorem
Take a lopsided pile of household incomes, grab `5` at random and average them, then repeat a thousand times. Those averages stack into a clean bell, and each extra draw buys less width than the one before.
- ▸Feed the machine a spiky, lopsided population and its sample averages still come out a smooth bell.
- ▸The averages pile far tighter than the population, and halving that width costs four times the data.
- ▸A bigger single sample never becomes a bell; only the pile of many samples' averages does.
Household incomes in a town pile up on the low end, most families clustered modestly, a few earners stretched far out into a long right tail. That histogram is nothing like a bell.
Now grab five households at random, average their incomes, and keep just that one average. Do it again with a fresh five, and again, a thousand times over. When you histogram those thousand averages, a clean symmetric bell appears, sitting right over the town's true average income. The lopsided raw pile never moved. The averaging is what built the bell: each average lets one household's high income partly cancel another's low one, so what survives, sample after sample, is the same tidy hump.