FALLACY OF AVERAGES

Fallacy of Averages - 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 E(XY) not= E(X)E(Y) – viz., the covariance is not zero. Another form of this fallacy is that E(f(X)) not= f(E(X)) (unless f is linear). In particular, suppose we have

P: max f(x; p): g(x; p) <= 0, where p is a vector of parameters with some uncertainty. The fallacy of averages suggests that it is a mistake to replace p with its expected value for at least 2 reasons: (1) members of p may be correlated, and (2) the average values of f and g need not equal the functional values at the mean.