The following comment was made by whuber in response to the question Which “mean” to use and when?
... in many cases the right mean to use is determined by the question we are trying to answer rather than by any mathematical structure in the data. A good example of this occurs in environmental risk assessment: regulatory authorities want to estimate a population's total exposure to contaminants over time. This requires an appropriately weighted arithmetic mean, even though environmental concentration data usually have a multiplicative structure. The geometric mean would be the wrong estimator or estimand.
This is a really important point and I am confident that I can prove to myself why it is true. However, it would be very helpful if someone could recommend an authoritative reference in the open literature that makes this point and explains why it is so.
