Harold Hotelling

When in different parts of a book there are passages from which the casual reader may obtain two different ideas of what the book is proving, and when one version of the thesis is interesting but false and the other true but trivial, it becomes the duty of a reviewer to give warning at least against the false version. My review of The Triumph of Mediocrity in Business was chiefly devoted to warning readers not to conclude that business firms have a tendency to become mediocre, or that the mediocre type of business tends with the passage of time to become increasingly representative or triumphant. That such a warning was needed is suggested by the title of the book and by various passages in it, and confirmed by the opinions of several eminent economists and statisticians who have taken the trouble to write or speak about the matter.

It is now clear that a tendency to stability or mediocrity of the kind which I showed was unproven, was not what the author intended to prove, and that a sufficiently careful reader would not be misled. But the thesis of the book, when correctly interpreted, is essentially trivial.

Consider a statistical variate x whose variance does not change from year to year, but for which there is a correlation r between successive values for the same individual. Let the individuals be grouped so that in a certain year all those in a group have values of x within a narrow range. Then among the mean values in these groups, the variance (calculated with the group frequencies as weights) will in the next year be less than that in the first year, in a ratio of which the mean value for linear regression and fine grouping is r², but in any case is η², less than unity. This theorem is proved by simple mathematics. It is illustrated by genetic, astronomical, physical, sociological and other phenomena. To “prove” such a mathematical result by a costly and prolonged numerical study of many kinds of business profit and expense ratios is analogous to proving the multiplication table by arranging elephants in rows and columns, and then doing the same for numerous other kinds of animals. The performance, though perhaps entertaining, and having a certain pedagogical value, is not an important contribution either to zoology or to mathematics.

Harold Hotelling, letter to the editor, Journal of the American Statistical Association, vol. 29, no. 186 (June, 1934),  pp. 198-199