The central limit theorem

Average lopsided dice and watch a bell curve appear out of nowhere.

Advanced: show a distribution where the theorem breaks

Numbers in each average5

n: how many draws get averaged into one dot.

Averages collected400

One draw at a time looks like this

Averages drawn400

Average of the averages5.37true mean 5.35

Spread of averages0.585theory σ/√n = 0.605

Skew of averages-0.87source skew -2.11

Each bar counts averages of 5 draws. Dashed: the bell curve the theorem predicts.

Single draws from the loaded die look nothing like a bell. Average 5 of them, though, and 400 averages already pile into a curve with spread 0.585, close to the predicted 0.605. The lean of the source fades as -2.11 ÷ √5 = -0.94.

How fast the lean disappears

Skew of the average as n grows. Lopsided sources need far more draws before a bell curve is a fair description, which is why rare-event risks are hard to model.

Try this

Pick the rare jackpot and set n to 1, then 10, then 100. When does the bell curve really fit?
Quadruple n from 5 to 20. The spread of the averages should halve, because of the √n.
Turn on the advanced toggle and raise n all the way. Does the Cauchy average ever settle?

The math

\bar{X}_n = \frac{1}{n}\sum_{i=1}^{n} X_i \;\approx\; \mathcal{N}\!\left(\mu,\ \frac{\sigma^2}{n}\right)

$\bar{X}_n$: the average of n independent draws
$\mu$: the true mean of one draw
$\sigma$: the standard deviation of one draw
$\sigma/\sqrt{n}$: standard error: how much averages wobble

Ask the Loop about this Next: Skill or luck? Bayesian updating