Alex's Adventures in Numberland - Alex Bellos [153]
In other words, if there are two rows of pins in the quincunx and we introduce lots of balls into the machine, the law of large numbers says that the balls will fall along the bottom such as to approximate the ratio 1:2:1.
If there are three rows, they will fall in the ratio 1:3:3:1.
If there are four rows, they will fall in the ratio 1:4:6:4:1.
If I carried on working out probabilities, a ten-row quincunx will produce balls falling in the ratio 1:10:45:120:210:252:210:120:45:10:1.
Plotting these numbers gives us the first of the shapes below. The shape becomes even more familiar the more rows we include. Also below are the results for 100 and 1000 rows as bar charts. (Note that only the middle sections of these two charts are shown since the values to the left and right are too small to see.)
So how does this pinball game relate to what goes on in the real world? Imagine that each row of the quincunx is a random variable that will create an error in measurement. Either it will add a small amount to the correct measurement or it will subtract a small amount. In the case of Galileo and his telescope, one row of pins could represent the temperature of the equipment, another could represent whether there is a thermal front passing through, and another could represent the pollution in the air. Each variable contributes an error either one way or the other, just as in the quincunx the ball will bounce left or right. In any measurement there may be many millions of unobservable random errors – their combined errors, however, will give measurements that are distributed like a bell curve.
If the characteristics of a population are normally distributed, in other words are clustered around an average in the shape of a bell curve, and if the bell curve is produced through random error, then, Quételet argued, the variation in human characteristics can be seen as errors from a paradigm. He called this paradigm l’homme moyen, or ‘the average man’. Populations, he said, were made up of deviations from this prototype. In Quételet’s mind, being average was something to aspire to since it was a way of keeping society in check – deviations from the average, he wrote, led to ‘ugliness in body as well as vice in morals’. Even though the concept of l’homme moyen never gained acceptance in science, its use filtered down to society at large. We often talk about morality or taste in terms of what an average representative of a population might think or feel about it: such as what is seen as acceptable ‘in the eyes of the average man’.
Whereas Quételet extolled averageness, Galton looked down on it. Galton, as I mentioned before, saw that exam results were normally distributed. Most people scobout average, while a few got very high marks and a few very low.
Galton, incidentally, was himself from a very above-average family. His first cousin was Charles Darwin, and the two men corresponded regularly about their scientific ideas. About a decade after Darwin published On the Origin of Species, which set out the theory of natural selection, Galton started to theorize on how human evolution itself could be guided. He was interested in the heritability of cleverness and wondered how it might be possible to improve the overall intelligence of a population. He wanted to shift the bell curve to the right. To this end Galton suggested a new field of study about the ‘cultivation of race’, or improving the intellectual stock of a population through breeding. He had thought to call his new science viticulture, from the Latin vita, ‘life’, but eventually settled on eugenics, from the Greek eu, good, and genos, birth. (The usual meaning of ‘viticulture’, grape cultivation, comes from vitis, Latin for ‘vine’, and dates from around the same time.) Even though many liberal intellectuals