It seems like taking the mean of a sample is not controversial. However, it could be the wrong thing to do. Consider this neat example from

D. J. Hand, “Deconstructing statistical questions,” J. Royal Statistical Society A (Statistics in Society), vol. 157, no. 3, pp. 317-356, 1994.

An English researcher and French researcher both test two cars of two types to determine which type is the more fuel efficient. One researcher measures miles per gallon, and the other gallons per mile. The following data are collected:

The English researcher finds the average miles per gallon of type 1 cars is greater than that of type 2, so they conclude type 1 is more fuel efficient. However, the French researcher finds the average gallons per mile of type 2 cars is less than that of type 1, so they conclude type 2 is more fuel efficient.

Who is right??

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Hello. I stumbled across your blog and enjoyed this post. The interesting question is not which researcher is correct, but rather why each may be wrong. The most obvious critique may be the failure to clearly define the figure of merit. While each mean is monotonic in the underlying measurement, the nonlinear 1/x relationship creates the possibility of scale-dependent ordering. Supposing we really meant to ask which car we should buy to minimize the cost of commuting, we might be tempted to adopt the French results. But there is a bigger problem. Most of us will not buy a fleet of cars. We must choose one. And we cannot purchase the “average car” from the dealer. So a better figure of merit would consider the variability in each population by asking which category offers greater confidence that our operating cost will not exceed a given budget threshold. Two samples per category does not offer very tight bounds, of course. Even with exhaustive sampling, we are gambling. Lemons happen. Thanks for the diversion!

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