Drop a thousand balls through a board of pegs, letting each one bounce left or right by pure chance. You don't design the bell curve — it builds itself, every single time.
A quincunx (Galton board) is a triangular array of pegs. A ball dropped in at the top hits a peg, bounces left or right with equal odds, hits another, bounces again — and after n rows, lands in one of n+1 bins at the bottom.
It's the most vivid demonstration of the Central Limit Theorem there is — random small steps adding up to a predictable bell shape. Quality control, polling margins, measurement error all rest on the same idea.
0 balls dropped through 9 rows of pegs. The dashed line is the theoretical normal curve — the histogram converges to it as you drop more.
Why a bell curve and not a flat pile?
Most balls take a mix of lefts and rights that roughly cancels, so they land near the middle. To land in an edge bin a ball must go *all lefts* or *all rights* — very unlikely. Many paths lead to the centre; only one leads to each edge. That imbalance is exactly the binomial distribution.
The probability of landing in bin k is C(n, k) / 2ⁿ — a Binomial(n, ½) distribution.
With 4 rows of pegs, how many paths lead a ball to the middle bin (bin 2)?
Drop 1,000 balls through 10 rows. Roughly how many land in the centre bin?
P(centre) = C(10, 5)/2¹⁰ = 252/1024 ≈ 0.246. So about 246 balls — by far the tallest column.
The board doesn't 'aim' for the middle. Each peg is a fair 50/50. The clustering is purely combinatorial — there are simply more ways to end up near the centre than at the edges.
Sir Francis Galton built the original device in the 1870s to show how heredity 'regresses toward the mean'. The same machine is sold today as a desk toy called a 'bean machine'.
- Each ball bounces left/right at every peg with equal probability.
- Final bin position follows a Binomial(n, ½) distribution.
- As the number of rows grows, that binomial becomes the normal (bell) curve — a live demo of the Central Limit Theorem.