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History of Statistics and the Quincunx

History of Statistics and the Quincunx

What is a Quincunx and where did it come from?

The Quincunx (pronounced quinn-cux) or bead board, as some call it, was developed by a mathematician named Galton in 1873 (complete history noted below). The device works by dropping a series of acrylic balls, or beads, through rows of precisely located pins. Each bead, as it hits a pin, has a 50-50 chance of falling to the left or right. Each bead then continues to fall over subsequent rows of pins and eventually lands in a slot or cell. The shape of the accumulated beads in the cells forms a pattern or what is statistically referred to as a bell shaped or normal distribution.

As any statistics student will tell you, a large number of data populations or industrial processes form a 'normal distribution', which is why the normal distribution is frequently used in statistics. A true statistician will also tell you that the bead distribution of a quincunx is actually a 'binomial distribution'. However, since the binomial and normal distributions are so much alike, we are safe in mathematically treating Quincunx distributions as if they are normally distributed. 

The real value of the quincunx is in being able to quickly simulate processes or tests that would be impractical to perform in real life. Some people actually refer to the quincunx ability to generate data as a “factory in box”. The quincunx performs three basic functions: 

  1. Generating real data 

  2. Visually displaying the data along with tolerances limits

  3. Being able to demonstrate changes in the data’s mean by shifting of the funnel to the right or left and changes in variability by adjusting the pin block. 

The following history of statistics describes how these principles were developed over time along with the invention of the quincunx.

A Brief History of Statistics and the Quincunx

Fundamental to the advancement of science is the need to measure physical phenomena and study the results. However, measurement alone is not enough since measurements are susceptible to comparison. Inherent problems with comparisons involve measurement accuracy, a way of measuring and expressing uncertainty, and some qualification of the inferential statements derived from the measurements. The need to answer these questions over the ages has birthed what is commonly known today as the field of statistics. 

Stephen M Stigler, wrote in his book The History of Statistics - The Measurement of Uncertainty before 1900 (pg 1) that: 

“modern statistics provides a quantitative technology for empirical science; it is a logic and methodology for the measurement of uncertainty and for an examination of the consequences of that uncertainty in the planning and interpretation of experimentation and observation.” 

Stigler’s book goes on to describe many of the early contributors to the field of statistics like Andrien Marie Legendre (1752-1833), who discovered the least squares principle, Pierre Simon Laplace, (1749-1827), who significantly contributed to the field of probability, physics, astronomy, and what math students will recognize by name as the Laplace transform in linear equations. Other notables include Abraham DeMoivre (1667-1754), Jacob Bernoulli (1654-1705), Carl Friedrich Gauss (1777-1855), and Adolphe Quetelet (1796-1874), who was first to begin the analysis of fitting normal curves to social data. 

In the 1880’s there was a notable change in the intellectual climate, as a series of remarkable men constructed an empirical and conceptual statistical methodology that provided a surrogate for experimental control. This methodology in effect dissipated the fog that had impeded statistical progress for a century. The three men responsible were Francis Galton (1822-1911), Ysidro Edgeworth (1845-1926), and Karl Pearson (1857-1936). These three men came from the divergent fields of anthropology, economics, and philosophy of science, but together created a statistical revolution. While each of these men had their peculiar strengths and weaknesses in developing and presenting this new methodology, the focus hereafter will be on Francis Galton who invented the quincunx as a means of communicating his new discoveries to a scholarly audience.

It seems that Galton had come across Adolphe Quetelet’s research and was inspired by his work of applying normal curves to social data. Galton’s continued research in this area resulted in him including Quetelet’s work in the appendix of his February 27, 1874 presentation on the “law of deviation from an average,” to the Royal Institution. Galton’s subject involved the effect of heredity on the height of individuals in Great Britain. Galton created the quincunx device with a dual purpose. For one he utilized the device for generating data to study and prove his theories. Second, he planned on using the device to display statistical distributions in support of his law of deviation from an average. 

Galton’s first quincunx, was built for him by a couple of scientists named Tisley and Spiller in 1873 and consisted of a framed board with a glass face and a stationary funnel at the top with 23 rows of pins. There was a hole at the top which allowed lead shot to be poured through the funnel and then to fall over the rows of pins with each shot having a 50-50 chance of falling to the right or left at each pin. Instead of individual slots like today’s quincunx design, Galton’s first quincunx had pockets that spanned 3-4 pin widths at the bottom of the quincunx for gathering small quantities of the fallen shot at the bottom of the quincunx. Pockets in the center of the quincunx had the most shot, while pockets near the edges had the least. Galton’s first quincunx and his design notes are still on display today at the Galton Laboratories in London England. (The Galton Laboratory is part of the Department of Genetics and Biometry at the University College London. Dr. J. S. Jones is the Department Head. ) The photographs below are of Dr. 

Gary Pittman’s visit to the Laboratories, pictured with the quincunx and June Rathbone, of the University. The second photograph is a close up of one of Galton’s actual quincunx. 

Galton built another quincunx in 1877. This was a two stage quincunx that illustrated that if a single pocket of shot from his first quincunx design, were to then flow over a second set of pins, that the resulting distribution would also be a normal shaped distribution. Thus Galton was first to demonstrate the fact that one normal distribution added or superimposed onto a second normal distribution is also normal. This principle was previously known by Laplace in its binomial form, but Galton’s work was new for the normal distribution. This discovery was perhaps the most significant breakthrough in statistics in the last half of the nineteenth century according to the previously mentioned author Stephen Stigler (page 281). The concept freed Galton of the restrictions from his naive error of data that was a mixture of very different populations. Following are some design sketches obtained during the same Dr. Pittman visit as noted on the Galton Laboratory website.



(www.galtoninstitute.org.uk/Newsletters )

Francis Galton was a gentleman scientist and had also a studied medicine at Cambridge. He went on to discover and refine many statistical tools like medians and percentiles as a way of characterizing distributions and reducing nonmetric data to a metric scale. He was also responsible for first discovering the principle of linear regression. Later in his life he would discover and validate the statistical basis of using fingerprints to uniquely identify individuals, much like DNA has been statistically validated today. While beyond the scope of this short paper, it is interesting to note that Francis Edgeworth and Carl Pearson would later present Galton’s work to many other scientists and provide even more applications of Galton’s principles to fields like astronomy. While Galton discovered the principles, the latter two really advanced them and presented them to the science community in various parts of the world.


Quincunx devices have continued to evolve over time. Today several science and history museums have large quincunx boards on display. Small stationary funnel quincunx boards, similar to Galton’s device have also been built and used in educational institutions. In the 1950’s an individual named Walter Kochel altered the quincunx design to become a small self contained unit. These units were sold under the name Lightning Calculator and manufactured in the inventor’s personal wood shop in St Petersburg, Florida. During the second wave of SPC in the 1980’s Jim Warren acquired theLightning Calculator name and adapted previous quincunx designs to create the modern day quincunx board while still maintaining the Lightning Calculator name.

Quincunx design and manufacturing techniques have significantly improved since the 1873 days of Tisley and Spiller in England. Todays boards are made of quarter sawn hard wood and machined on CNC routers. Pin blocks are precisely drilled and have adjustable pin patterns. The funnels are also now moveable and self locking. One thing that has not changed however is the fact that practitioners still refer to the quincunx as a “factory in a box” and use it to generate data the same way that Galton did with his quincunx.

One more thing... where did the name “quincunx” come from? Quincunx is the name of a Roman coin that had five indentions on its face. Four points were stamped on the coin in a square design with the fifth point in the center of the square.

This pattern has also been used over time in the agriculture world as a pattern for planting plant trees. Since this was the name of the pin pattern according to Galton’s work (p. 64 of Natural Inheritance, by Francis Galton) the device has been known as a quincunx (pronounced like quinn-cux) ever since. It interesting to note that Galton originally call the device an “Instrument to illustrate the principle of the Law of error or Dispersion.” .... thank goodness that “quincunx” became the short name for this device.

More information can be found about Francis Galton at http://www.galton.org.  

Copyright 2014
Lightning Calculator
by Jim Warren



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