How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

Hotelling was totally devoted to research and the generation of knowledge, and in Secrist he may have seen something of a kindred soul. “The labor of compilation and of direct collection of data,” he wrote sympathetically, “must have been gigantic.” Then the hammer drops. The triumph of mediocrity observed by Secrist, Hotelling points out, is more or less automatic whenever we study a variable that’s affected by both stable factors and the influence of chance. Secrist’s hundreds of tables and graphs “prove nothing more than that the ratios in question have a tendency to wander about.” The result of Secrist’s exhaustive investigation is “mathematically obvious from general considerations, and does not need the vast accumulation of data adduced to prove it.” Hotelling drives his point home with a single, decisive observation. Secrist believed the regression to mediocrity resulted from the corrosive effect of competitive forces over time; competition was what caused the top stores in 1916 to be hardly above average in 1922. But what happens if you select the stores with the highest performance in 1922? As in Galton’s analysis, these stores are likely to have been both lucky and good. If you turn back the clock to 1916, whatever intrinsic good management they possess should still be in force, but their luck may be totally different. Those stores will typically be closer to mediocre in 1916 than in 1922. In other words, if regression to the mean were, as Secrist thought, the na...