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The Galton Box

History of Statistics, History of Science and Technology

The Galton Box (also known as a bean machine or quincunx), was invented by Sir Francis Galton in the 19^th century. Its purpose is to demonstrate the central limit theorem and the binomial and normal distributions.

As a ball falls, it will bounce either left or right when it hits a pin. If there are N rows of pins, then the ball would bounce N times. The position of the bin the ball falls in, would have a binomial distribution where:

binomial distribution formula

where:

x is the bin position e.g. x=0 could be treated as the left-most bin, and x=N could be treated as the right-most bin.
P is the probability of x
p is the probability of bouncing right (if x=0 represents the left-most bin). In an unbiased machine: p = 0.5.
N is the number of rows of pins i.e. the number of times a ball bounces.

Note that for the positions to conform to a binomial distribution, the number of bins should equal N+1.

If the number of rows of pins is large enough, this would approximate a normal distribution due to the central limit theorem.