About the Galton Board | Chance, Pascal's Triangle, and the Bell Curve

Francis Galton's machine

Francis Galton, a Victorian scientist with a passion for heredity and statistics, built the first board in the 1870s. He described it in his book Natural Inheritance in 1889. He called it a quincunx, after the pattern of five dots on a die, which the staggered pegs repeat.

One bounce at a time

At every peg a ball goes left or right with equal chance, like a coin toss. After n rows it has made n such choices. Its bin is the number of times it went right. The order does not matter: right then left ends in the same place as left then right.

Pascal's triangle counts the paths

Write 1 at the top peg. At every other peg, add the two numbers above it. That is Pascal's triangle, and each number counts the paths that reach that peg. The last row gives the paths into each bin. For 12 rows it runs 1, 12, 66, 220, 495, 792, 924 and back down.

The binomial distribution

Divide the paths into a bin by all 2ⁿ paths, and you have the chance of landing there. That is the binomial distribution. On a fair board the mean is n/2 and the standard deviation is √n/2. With a bias p the mean becomes n·p and the spread √(n·p·(1−p)).

From bins to a bell curve

As the rows increase, the bars of the binomial distribution trace out the smooth normal curve, the bell curve. Abraham de Moivre showed this in 1733. It is a first case of the central limit theorem: add up many small, independent random steps, and the total follows a bell curve.

Why the bins are never exact

Each ball is random, so the bins scatter around their expected heights. A bin that should hold about N balls is usually off by about √N. Drop 100 balls and the shape is rough. Drop 5000 and it hugs the curve.

Galton and regression to the mean

Galton used the board in his work on heredity. He noticed that very tall parents tend to have children who are tall, but closer to the average. He called this regression to the mean, an idea now used all across statistics.

Try it in PictorX

Open the Galton board and the balls start falling at once. Rows sets the number of bounces, Balls how many fall, and Bias the chance of going right. The bell curve shows the expected heights, and the numbers under the bins count the balls. Turn on Pascal's triangle to see the paths, then save a PNG.

Frequently asked questions

Who invented the Galton board?

Francis Galton built it in the 1870s and described it in 1889 in his book Natural Inheritance.

What is the difference between the binomial and the normal distribution?

The binomial distribution counts the successes in a fixed number of yes-or-no trials, like the right bounces on a board. The normal distribution is the smooth bell curve it approaches when the number of trials is large.

Why is it called a quincunx?

A quincunx is the pattern of five dots on a die: four in the corners and one in the middle. The staggered rows of pegs repeat that pattern.

Does a real Galton board give a perfect bell curve?

Not quite. Real balls hit each other and can bounce more than one step at a time, which widens the bell a little. The bounces in this simulation are independent, as on an ideal board.