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In the biological literature, the mean of the product of two or more random variables is frequently calculated from the product of their means. However, unless the variables are independent, an exceptional occurrence in biological systems, the two are not equivalent. Corresponding false assumptions commonly are made about ratios of means and various other functions of means. These assumptions are examples of perhaps the most common statistical fallacy in the biological literature, the fallacy of averages: the false assumption that the mean of a nonlinear function of several variables equals the function of the means of those variables. We provide the relationship between functions of means and means of functions for common functions of one variable (linear, reciprocal, and exponential functions), for two or more variables (product, ratio, sum), and for the product of allometric relationships.
Welsh et al. (Mon,) studied this question.