What does it mean to be average? : the miles per gallon versus gallons per mile paradox revisited

In the efficiency paradox, which was introduced by Hand (1994; J R Stat Soc A, 157, 317-356), two groups of engineers are in disagreement about the average fuel efficiency of a set of cars. One group measured efficiency on a miles per gallon scale, the other on a gallons per mile scale. In the present paper, I argue against an operationalistic explanation of the efficiency paradox, by showing that the paradox is neither the result of an ambiguously defined efficiency concept, nor the result of how fuel efficiency is measured (i.e., whether a miles per gallon, or gallons per mile scale is used). The actual paradox is that the two groups of engineers have asked different statistical questions, while using the same mathematical operation. The paradox results from the fact that fuel efficiency is a derived measure, like density and speed, for which end-to-end concatenation (i.e., addition) is not straightforward.