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The Galton Box

History of Statistics, History of Science and Technology

Bean machine

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

  • 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.


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