Geometric and weighted mean

[StephenG1CMZ]

(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.

I was intruiged by your mention of max and min as extreme examples of means. I realised that I really was not sure what "a mean" was. To try to understand the concept, I looked up several example means - and implemented many of them here.
http://www.hpmuseum.org/forum/thread-9852.html
I admit I still would not like to have to explain what "a mean" is.

As for justifying which mean to use for a given data set, I thought my IsMean procedure might be useful. Used properly, it can show which mean a tabloid newspaper has used in its headline. Used improperly, given the number you would like as your answer, it can identify which average you need to use (regardless of appropriateness).

By the way, if you have weighted data but your mean procedure does not implement weights, you can use my InvOCCUR procedure to turn Valuelist,Weightlist into Valuelist-with-repeated-values (I'm sure there must be a better name). Obviously, weights must be integer.
http://www.hpmuseum.org/forum/thread-9411.html