When to use geometric vs arithmetic mean?

Question

So I guess this isn't technically a code question, but it's something that I'm sure will come up for other folks as well as myself while writing code, so hopefully it's still a good one to post on SO.

The Google has directed me to plenty of nice lengthy explanations of when to use one or the other as regards financial numbers, and things like that.

But my particular context doesn't fit in, and I'm wondering if anyone here has some insight. I need to take a whole bunch of individual users' votes on how "good" a particular item is. I.e., some number of users each give a particular item a score between 0 and 10, and I want to report on what the 'typical' score is. What would be the intuitive reasons to report the geometric and/or arithmetic mean as the typical response?

Or, for that matter, would I be better off reporting the median instead?

I imagine there's some psychology involved in what the "best" method might be...

Anyway, there you have it.

Thanks!

Answer 1

you want the arithmetic mean. since you aren't measuring the average change in average or something.

Answer 2

Generally speaking, the arithmetic mean will suffice. It is much less computationally intensive than the geometric mean (which involves taking an n-th root).

As for the psychology involved, the geometric mean is never greater than the arithmetic mean, so arithmetic is the best choice if you'd prefer higher scores in general.

The median is most useful when the data set is relatively small and the chance of a massive outlier relatively high. Depending on how much precision these votes can take, the median can sometimes end up being a bit arbitrary.

If you really really want the most accurate answer possible, you could go for calculating the arithmetic-geomtric mean. However, this involved calculating both arithmetic and geometric means repeatedly, so it is very computationally intensive in comparison.