The Sampling Lab

The Galton board

Balls fall through rows of pins, each bounce a coin flip, and the bins fill into the binomial distribution.

Each ball meets a row of pins at every level and bounces right with probability p. Its bin counts its right bounces. The short horizontal ticks mark the exact binomial expectation for the balls so far: more balls only fill the bins in, while more rows of pins widen the pile. Change the rows and the tilt, drop 500 at once, and switch on the normal curve.

A Galton board (also called a bean machine or quincunx) is a vertical board with staggered rows of pins. Balls dropped in at the top bounce left or right at each pin and collect in bins at the bottom, where they pile into a bell-shaped curve. With 12 rows of pins, each bin’s chance is a binomial probability, and the centre bin collects about 23 % of the balls while each edge bin collects 1 in 4,096.

Francis Galton described the board in his 1889 book Natural Inheritance as a way to show how a regular, bell-shaped distribution arises from many small, independent and individually unpredictable deflections[1]. It is still the clearest physical picture of the binomial distribution, and of the central limit theorem behind it. documented

Why the middle fills first

A ball’s bin is its number of right bounces. With 12 rows there is exactly one path to the leftmost bin (left, left, left, …) and one to the rightmost, but 924 different paths end in the middle bin, because there are 924 ways to choose which 6 of the 12 pins send the ball right. On a fair board every path is equally likely, with probability 1/212 = 1/4,096, so the middle bin is 924 times as likely as an edge bin. The path counts are the numbers in row 12 of Pascal’s triangle, and the bin probabilities are the binomial distribution[2]:

P(bin k) = C(n, k) pk (1 − p)n − k(1)

With 12 rows and p = ½ the three central bins together take 61 % of the balls. Watch the first dozen balls land and the pile looks ragged; by a few hundred it hugs the ticks, which mark the exact expected count in each bin. established

How wide is the pile?

Each bounce adds a little randomness: a right bounce adds one to the bin number with probability p, and the variance of one bounce is p(1 − p). Variances of independent bounces add, so after n rows the variance is np(1 − p) and the standard deviation √(np(1 − p)). For 12 fair rows that is √3 = 1.73 bins; for 24 rows it is √6 = 2.45. Doubling the board makes the pile only √2 times wider, the same square-root law that governs the precision of averages in the chapter on the central limit theorem. Set the board to 12 rows, then 24, and read the spread under it.

Tilting the board

Set p to 0.2 and the balls drift left, and the pile leans: with 12 rows its skewness is 0.43. Add rows and the lean fades, to 0.31 at 24 rows, because the skewness of a binomial count, (1 − 2p)/√(np(1 − p)), shrinks as n grows. Switch on the normal curve to see the approximation improve with every row. Abraham de Moivre proved in 1733 that the binomial distribution approaches this curve for large n, the first appearance of the normal distribution in mathematics[3]. established

Galton’s board

Galton called his device the quincunx, after the staggered pattern of its pins, the arrangement of five points on the face of a die. He used it as an argument by analogy: if many small independent influences add up, the result is bell-shaped, and so, he thought, are the heights and other traits he measured in human populations[1][3]. Galton’s interest in heredity also led him to the idea of regression toward the mean, and the board became one of the standard illustrations of chance in statistics teaching. documented

For exact probabilities on any number of rows or any tilt, use the binomial calculator; for the Poisson distribution and the other shapes counting processes make, see the chapter on where distributions come from.

Sources

  1. Galton, F. (1889). Natural Inheritance. Macmillan, London (ch. V, the apparatus later called the Galton board). primary
  2. NIST/SEMATECH (2012). NIST/SEMATECH e-Handbook of Statistical Methods, §1.3.6.6.18 Binomial Distribution. National Institute of Standards and Technology. standard
  3. Stigler, S. M. (1986). The History of Statistics: The Measurement of Uncertainty before 1900. Belknap Press of Harvard University Press. ISBN 978-0-674-40341-3. scholarship Ch. 2 (de Moivre’s normal approximation, 1733), ch. 4 (Laplace and the central limit theorem), ch. 8 (Galton).