Geometric and weighted mean

[salvomic]

(12-24-2017 03:01 AM)AlexFekken Wrote:  Perhaps it is worth pointing out that *an* average is a relatively arbitrary way of "summarising" a bunch of numbers into a single number. The abstract definition that I learnt at uni even covers the two extreme cases of min() and max().

It is very common not to justifiy a particular choice of average (e.g. the arithmetic mean), other than by its ubiqitous implementation. If you want to get really serious and properly justify why you "should" use a certain choice of average, you are typically lead to consider cost functions. For example: minimize sum of squares (arithmetic mean), miminize expected walking distance to the first lift that arrives in the office (median), ...

In terms of relative size, these hold in general:

min <= harmonic <= geometric <= arithmo-geometric <= arithmetic <= max

I think not much can be said about the relative sizes of other common types as it depends heavily on the (skewness of the) distribution of the numbers. In fact, their size relative to the arithmetic mean may be used as an indicator of skewness.

yes Alex, definitely all depends after the data we want treat and after the problem type.
That said, the Prime has already good instruments to handle Statistics. Perhaps it is missed only geometric mean, that in some cases is more adequate than other types or of the median. It is not a great problem as we can make it with a little program; my friend would like only to have it as a command; I tried to help him with a program that acts like a command, so he can experiment among the various types and I'll suggest him also your right considerations about the correct use of means.
Thank you,
Salvo