Geometric and weighted mean

[AlexFekken]

(01-04-2018 03:55 PM)StephenG1CMZ Wrote:  I admit I still would not like to have to explain what "a mean" is.

Nice to see someone taking my feedback seriously and expanding on it.

From memory, the abstract definition of a mean (that I learned in a "captia selecta" math lecture) is that of any function of a "finite sequence" (i.e. a list) of numbers satifying:

1 - the mean is not less than all of the numbers
2 - the mean is not more than all of the numbers
3 - the mean is scale-invariant

Presumably the numbers had to be real (or at least from a totally ordered set) for 1 and 2 to make sense. I am not sure if there were further domain restrictions, e.g. all numbers > 0, in this abstract context, but since we covered the arithmetico-geometric mean I suppose there must have been. And "weights" apparently weren't in scope at all :-)

The 3rd condition means that if you multiply all numbers in the seqence by a fixed constant then you can get the new mean by multiplying the original by the same constant. This ensures that if the number represent physical quantities on an absolute scale (e.g. NOT degrees Celsius, Fahrenheit), then you will get the same mean regardless of the units used. So other than "staying within range" that was the only functional restriction on the definition of a "mean".

But of course abstract definitions are not intended to explain but to distil the essence of a concept (my favourite being that of a "topology" to represent "closeness" without talking about distance). I would say using cost functions is probably the best way to justify/explain the different types of means (as "best estimates" that mimimize them). For example (and in addition to the ones I already mentioned), if the cost of underestimating is much very bigger than that of overestimating, then max would be your best "mean".