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A very interesting piece by Howard Wainer in the latest American Scientist (May-June 2007) concerns dangerous equations, which he describes as falling into two classes:
- equations that are dangerous because we know them - they "may pose danger because the secrets within its bounds open doors behind which lies terrible peril," with E=mc2 the most obvious candidate
- equations that are dangerous because we don't know them - mot because there is no theory that has yet yielded these equations, but rather because they are not known by those who need to know them. This is especially true for policy makers that base their decision on mathematical models, and specifically statistical models.
Wainer's top choice for most dangerous statistical equation is due to Abraham de Moivre, who showed in 1730 that the standard error of the mean of a sample is the standard error of the mean of the population divided by the square root of the sample size. A significant prediction of this equation is that small sample sizes lead to large fluctuations in sample means. It is this simple statement:
small samples → large fluctuations in sample means,
that provides the biggest danger when not used, or not understood, by both policy makers and the average citizen.
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