# The Most Dangerous Equation

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1 The Most Dangerous Equation Ignorance of how sample size affects statistical variation has created havoc for nearly a millennium Howard Wainer Wliat constitutes a dangerous equation? There are two obvious interpretations: Some equations are dangerous if you know them, and others arc dangerous if you do not. The first category may pose danger because the secrets within its bounds open doors behind which lies terrible peril. The obvious winner in this is Einstein's iconic equation e = nic~, for it provides a measure of the enormous energy hidden within ordinary matter. Its destructive capability was recognized by Leo Szilard, who then instigated the sequence of!'. The Goldsmiths't!iiiiipLii f;p fi Figure 1. Trial of the pyx has been performed since 1150 A.D. In the trial, a sample of minted coins, say 100 at a time, is compared with a standard. Limits are set on the amount that the sample can be over- or underweight. In 1150, that amount was set at 1/400. Nearly 600 years later, in 1730, a French mathematician, Abraham de Moivre, showed Ihat the standard deviation does not increase in proportion to the sample. Instead, it is proportional to the square root of the sample size. Ignorance of de Moivre's equation has persisted to the present, as the author relates in five examples. This ignorance has proved costly enough that the author nominates de Moivre's formula as the most dangerous equation May-June 249

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