Fallacy of averages

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Imagine standing with one foot on a keg of ice and the other in a fire. Your average body temperature may be fine, but the extremes will kill you! The fallacy of averages is the fallacious results you may get when replacing data with their expected values. Formally, the fallacy is stated as LaTeX: E(XY) \ne E(X)E(Y) – viz., the covariance is not zero. Another form of this fallacy is that LaTeX: E(f(X)) \ne f(E(X)) (unless LaTeX: f is linear). In particular, suppose we have

LaTeX: \max f(x; p): g(x; p) \le 0,

where LaTeX: p is a vector of parameters with some uncertainty. The fallacy of averages suggests that it is a mistake to replace LaTeX: p with its expected value for at least 2 reasons: (1) members of LaTeX: p may be correlated, and (2) the average values of LaTeX: f and LaTeX: g need not equal the functional values at the mean.

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