Galton machine

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Copyright 1996, 2000, Denis Cousineau


Francis Galton was a British physicist and researcher. He is most famous for creating the correlation method. He also published papers in mathematics, including the first paper on the Lognormal distribution and papers in experimental psychology, favoring nephrology (readings of the bumps on the head) and eugenic (sterilizing people with low I.Q.). These last researches are very questionable.

Galton first proposed a simple machine to generate data that are normally distributed. This device is a simple application of the theory of errors, first proposed by Gauss. Gauss argued (and demonstrated) that under the following conditions, a precise bell-shaped distribution should be obtained: the quantities measured (the scores) are the results of (i) a large number of factors, (ii) each factor being either favorable of unfavorable (iii) in a proportion of about 50/50.

For example, suppose that we are interested in the height of people. We know that a lot of factors can influence the height, including genetic factors, nutritional factors, general health, trauma in childhood, etc. The exact number of factors is unknown, but definitely large. Some are favorable for taller persons, such as a good diet, others are unfavorable, such as trauma during teen ages.

This general theory was called by Gauss the theory of error, suggesting that each factor has a more or less precise impact on the score. Under these very general conditions, Gauss showed in 1812 that the probability of obtaining a certain scores, call it x, should be distributed around the average m with the following equation:

This equation, depicting a bell-shape curve, is now famous, lying at the heart of many statistical tests, such as the t-test, the analyses of variance, etc. It is often called the "Normal" distribution (although there is nothing normal about it), less frequently, the Gaussian distribution.

At a time where computer simulations were not even a dream, it was difficult to demonstrate that the equation was true, whatever the system, as long as the above assumptions were true. 

Galton proposed a very simple device which meets them, and therefore, which should generate normally distributed scores. It is very simple: simply have a surface lift at an angle of approximately 45 degrees. Places hundreds of nails on it so that a marble cannot fall on it without hitting many of them. At the bottom, install gutters so that you know where the marble ended its course. That's it.

Each nail is a "factor" that will influence the final position of the marble. When hitting a nail, the marble can continue on the right or the left with near 50% of chance. Going to the left will favor 'small' final position whereas the opposite will favor 'large' resting position. Of course, small and large is arbitrary here. These two satisfies the above assumptions (ii) and (iii). Because there are many nails, assumption (i) is also satisfied. Because all the assumptions are true of this device, the conclusion ought to be true as well, that the positions of the marbles will be normally distributed.

I decided to built this device, called a Galton machine in honor of its creator. It is not exactly a machine though since the essential parts (the numerous nails) don't move. To make it, I made the largest panel possible with technic beams. Then, I installed black pegs (3 units long) to act as nails:

in the holes, close to each others but with enough space so that the marbles can circulate freely. I placed gutter at the bottom. The walls between the gutters were labeled (arbitrarily) -10 to +10, so that a marble falling in the leftmost gutter was said to have fallen in position -6.5 (between the -7 wall and the -6 wall). Finally, on top of this, I install a marble distributor, a system that pull from a pool containing marbles only one at a time (this was maybe the most difficult part of the machine). The resulting construction is shown in the following two clickable pictures:

front.jpg (302769 bytes) side.jpg (356236 bytes)

The following plots shows two tests of the Galton machine. A total of 80 marbles were placed in the pool. After the run, the graphs shows how many marbles ended in each of the gutter:

As seen, the results do not resemble exactly a bell-shape curve. One reason is that the total number of marble is small (80). Distributions are generally stable, but depends amply on random factors. Thus, if the machine could be run with, say, 200 marbles, the results would look much more like a bell, and with 800 marbles, the obtained distribution would be almost identical to the curve obtained from the equation. Of course, the gutters on my own Galton machine cannot contain so many marbles, so a more complete test will have to wait...