The Mathematics of Statistics

For most of us, mathematics is a pain in the neck, and we avoid it whenever we can. We hire an expert to figure our tax. We trust the clerk to make the right change. We buy meat weighed on a computing scale and gasoline from a computing pump. We learned algebra in school, but we've forgotten it! What wonder that most of us neither use nor understand the mathematics of statistics, which deals with general tendencies and random differences. We'd be glad to forget this, too, if we'd ever learned it. Or perhaps we wouldn't forget it, for there's a good likelihood that we'd find frequent occasion to use it, which is certainly not true of algebra. Statistics deal with chances. It attempts to tell, from looking at a few patients, what one could expect in 100 or 10,000—from making a few measurements, what they would be if one hundred times repeated. Chances indicate the number of times one would encounter this or that in a large population. For me chances have no meaning unless there exists in reality, or in my imagination, a large population or a large number of repetitions of the observation or the measurement in question. Ten per cent mortality means that 100 out of 1,000 die. But if a single person, in complete isolation, has the same disease, he either lives or dies, 100 per cent not 10 per cent. Since the physician deals with patients as individuals one at a time, statistical statements must always have a degree of unreality for him. To make them applicable to the real situation, he must in his imagination apply the same logic, arriving at the same advice, for many similar patients in the future. His aim is to end up his career with the largest number of old patients still 100 per cent living and the fewest 100 per cent dead. The mathematics of statistics has been worked on by some of the finest brains of the last century. The theorems used in standard statistical calculations are an honest part of mathematics—are rigorously proved. This seems a paradox when we consider the uncertainty, or rather the random scattering of the data dealt with. The concept of randomness is actually of the essence. The most frequent blunder that destroys the trustworthiness of statistical evidence is non-random sampling. It is so natural to operate on those patients who are most promising for cure. A high percentage of five-year cures put forward as proving that the operation is a good one may only prove that the surgeon was timid. The statistical conclusion was, in this imaginary instance, actually inserted into the data by the act of the investigator. That he did so innocently does not make the conclusions any more valid. When it is willful, and the stuff put into the sample is gold, we call it “salting.” There are other common blunders. One is to calculate everything in per cent.