Drunkard's Walk - Leonard Mlodinow [70]
In judging the meaning of their measurements, scientists in the eighteenth and nineteenth centuries faced the same issues as the skeptical oenophile. For if a group of researchers makes a series of observations, the results will almost always differ. One astronomer might suffer adverse atmospheric conditions; another might be jostled by a breeze; a third might have just returned from a Madeira tasting with William James. In 1838 the mathematician and astronomer F. W. Bessel categorized eleven classes of random errors that occur in every telescopic observation. Even if a single astronomer makes repeated measurements, variables such as unreliable eyesight or the effect of temperature on the apparatus will cause the observations to vary. And so astronomers must understand how, given a series of discrepant measurements, they can determine a body’s true position. But just because oenophiles and scientists share a problem, it doesn’t mean they can share its solution. Can we identify general characteristics of random error, or does the character of random error depend on the context?
One of the first to imply that diverse sets of measurements share common characteristics was Jakob Bernoulli’s nephew Daniel. In 1777 he likened the random errors in astronomical observation to the deviations in the flight of an archer’s arrows. In both cases, he reasoned, the target—true value of the measured quantity, or the bull’s-eye—should lie somewhere near the center, and the observed results should be bunched around it, with more reaching the inner bands and fewer falling farther from the mark. The law he proposed to describe the distribution did not prove to be the correct one, but what is important is the insight that the distribution of an archer’s errors might mirror the distribution of errors in astronomical observations.
That the distribution of errors follows some universal law, sometimes called the error law, is the central precept on which the theory of measurement is based. Its magical implication is that, given that certain very common conditions are satisfied, any determination of a true value based on measured values can be solved employing a single mathematical analysis. When such a universal law is employed, the problem of determining the true position of a heavenly body based on a set of astronomers’ measurements is equivalent to that of determining the position of a bull’s-eye given only the arrow holes or a wine’s “quality” given a series of ratings. That is the reason mathematical statistics is a coherent subject rather than merely a bag of tricks: whether your repeated measurements are aimed at determining the position of Jupiter at 4 A.M. on Christmas Day or the weight of a loaf of raisin bread coming off an assembly line, the distribution of errors is the same.
This doesn’t mean random error is the only kind of error that can affect measurement. If half a group of wine critics liked only red wines and the other half only white wines but they all otherwise agreed perfectly (and were perfectly consistent), then the ratings earned by a particular wine would not follow the error law but instead would consist of two sharp peaks, one due to the red wine lovers and one due to the white wine lovers. But even in situations where the applicability of the law may not be obvious, from the point spreads of pro football games25