Monday 2 January 2012

The Quincunx

The Quincunx was designed by Sir Francis Galton (1911-1911)  as a physical simulation device to show how a Gaussian distribution can arise from the repeated application of a random choice of 50/50 probabilities. This is the central limit theorem of statistics. The board starts with 1 pin on the first row, 2 pins on the second row, 3 pins on the third row, and so on. Multiple balls are then dropped onto the top pin. The ball must fall to the left or right, with roughly 50/50 chance. As the balls fall through the multiple layers of pins to the bottom, they will land into bins which are placed below the last row of pins.

If there are a large number of balls used then when a count is made of the number of balls in each bin, one notices that there are more balls in the center bins than there are in the outer bins. Mathematically, we get an approximation to a normal, or Gaussian,  distribution.

The word Quincunx itself is fantastic (see definition here at Collins Dictionary Online). A quincunx was originally a coin issued by the Roman Republic c. 211–200 BC, whose value was five twelfths (quinque + uncia) of an As, the Roman standard bronze coin.  On the Roman quincunx coins, the value was sometimes indicated by a pattern of five dots or pellets. Wikipedia has a good list of where the Quincunx pattern has been used.  

Galtons original Quincunx is housed in the Galton Institute in London - HERE.

There is a Java simulation of the Quincunx HERE. Below is an example of a simulation from it.

Below is an image of Galtons original Quincunx from the book HERE.