Geometric and weighted mean

[AlexFekken]

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.