Drunkard's Walk - Leonard Mlodinow [74]
THE RECOGNITION that the normal distribution describes the distribution of measurement error came decades after De Moivre’s work, by that fellow whose name is sometimes attached to the bell curve, the German mathematician Carl Friedrich Gauss. It was while working on the problem of planetary motion that Gauss came to that realization, at least regarding astronomical measurements. Gauss’s “proof,” however, was, by his own later admission, invalid.31 Moreover, its far-reaching consequences also eluded him. And so he slipped the law inconspicuously into a section at the end of a book called The Theory of the Motion of Heavenly Bodies Moving about the Sun in Conic Sections. There it may well have died, just another in the growing pile of abandoned proposals for the error law.
It was Laplace who plucked the normal distribution from obscurity. He encountered Gauss’s work in 1810, soon after he had read a memoir to the Académie des Sciences proving a theorem called the central limit theorem, which says that the probability that the sum of a large number of independent random factors will take on any given value is distributed according to the normal distribution. For example, suppose you bake 100 loaves of bread, each time following a recipe that is meant to produce a loaf weighing 1,000 grams. By chance you will sometimes add a bit more or a bit less flour or milk, or a bit more or less moisture may escape in the oven. If in the end each of a myriad of possible causes adds or subtracts a few grams, the central limit theorem says that the weight of your loaves will vary according to the normal distribution. Upon reading Gauss’s work, Laplace immediately realized that he could use it to improve his own and that his work could provide a better argument than Gauss’s to support the notion that the normal distribution is indeed the error law. Laplace rushed to press a short sequel to his memoir on the theorem. Today the central limit theorem and the law of large numbers are the two most famous results of the theory of randomness.
To illustrate how the central limit theorem explains why the normal distribution is the correct error law, let’s reconsider Daniel Bernoulli’s example of the archer. I played the role of the archer one night after a pleasant interlude of wine and adult company, when my younger son, Nicolai, handed me a bow and arrow and dared me to shoot an apple off his head. The arrow had a soft foam tip, but still it seemed reasonable to conduct an analysis of my possible errors and their likelihood. For obvious reasons I was mainly concerned with vertical errors. A simple model of the errors is this: Each random factor—say, a sighting error, the effect of air currents, and so on—would throw my shot vertically off target, either high or low, with equal probability. My total error in aim would then be the sum of my errors. If I was lucky, about half the component errors would deflect the arrow upward and half downward, and my shot would end up right on target. If I was unlucky (or, more to the point, if my son was unlucky), the errors would all fall one way and my aim would be far off, either high or low. The relevant question was, how likely was it that the errors would cancel each other, or that they would add up to their maximum, or that they would take any other value in between? But that was just a Bernoulli process—like tossing coins and asking how likely it is that the tosses will result in a certain number of heads. The answer is described by Pascal’s triangle or, if many trials are involved, by the normal distribution. And that, in this case, is precisely what the central limit theorem tells us. (As it turned out, I