Geometric vs. arithmetic mean I'm not sure if this is the best place for this question, but when averaging data, how do you know if the geometric or arithmetic mean is more appropriate for taking the average? Do you need to know something a priori about the data set to decide? For my particular problem, I am sampling the data in a log space and wish to average several data sets that should follow the same trends.
 A: As with many things, it depends on what you want to get out of it.
The neat thing about the arithmetic average is that if you have one, and you know the total number of values, you can multiply the two and get the total sum. If that property is important, you use the arithmetic average.
Something neat about the geometric average is that it respects compounding, making it appropriate for financial comparisons of total wealth under various courses of action.
But I don't know of a strict rule, other than that I can list many examples where one is appropriate and the other is not. But sometimes it's not even that clear.
A: There are lots of possible aggregations besides the arithmetic and geometric mean, popular other choices are e.g. the harmonic mean or the generalized mean, or, as a robust alternative, the median.
For you, it might be of relevance that the logarithm of the geometric mean is the arithmetic mean of the logarithms.
You might want to always consider at least the three Pythagorean means, harmonic ($HM$), geometric ($GM$), and arithmetic ($AM$) mean. Note, that their values on any set $S$ of numbers always satisfy
$$
HM(S) \le GM(S) \le AM(S).
$$
Intuitively, small values have more weight in the harmonic and geometric mean, so if a value goes to zero, it draws the mean to zero, too, which is not the case with the arithmetic mean.
A nice property of $GM$ is that the geometric mean of ratios is the ratio of the geometric means:
$$
\sqrt[n]{\frac{x_1}{y_1}\ldots\frac{x_n}{y_n}} = \frac{\sqrt[n]{x_1\cdots x_n}}{\sqrt[n]{y_1\cdots y_n}}.
$$
