# Arithmetic vs Geometric Mean

There are verbal suggestions everywhere on when one should use a geometric average or when an arithmetic average should be preferred, but I can't find any formal statistical treatment of this question. Is it possible to formally test which one of these averages should be used for a particular sample?

• What do you mean and want by formally test? A numeric example with verbal moral won't suit? – ttnphns Mar 13 '14 at 5:58
• No, there aren't any formal tests of this. And it isn't the particular sample that matters as much as what the variable is and what you are trying to get. There are ways to show that the arithmetic mean gives wrong results in some cases. – Peter Flom - Reinstate Monica Mar 13 '14 at 9:55
• Thanks for the response. Shouldn't distribution of the data have a bearing on which kind of a mean should be used? For example, is it wrong to sugegst using arithmetic mean if the data is normally distributed and geometric mean if its log is normally distributed? If such claims can be made, why can't we have a statistical test for it? – user41838 Mar 14 '14 at 3:50
• This has come up before. – Dimitriy V. Masterov Mar 16 '14 at 5:13

## 1 Answer

Such decisions are not normally made on the basis of testing, but on an understanding of the variables, the circumstances and the needs of the analysis.

For example, we would need to consider when it is more meaningful to us to use a geometric mean and when it is more meaningful to use an arithmetic mean.

If I am interested in a population mean, generally speaking I probably want to consider a sample arithmetic mean. However, if I have particular distributional assumptions, it's possible a geometric mean may (perhaps after some transformation, or in combination with other quantities) lead me to a better estimate of the population mean than the arithmetic mean (e.g. in a lognormal distribution, the geometric mean might be part of the calculation of a better estimate of the population mean than the arithmetic mean).

You seem to be suggesting that we can infer the best estimator from the data, as if we perform two stages of estimation, first of the distribution, and then from that choose a good estimator.

But in fact it's not at all clear that this is generally a good approach to estimation, except perhaps in certain classes of estimation problems, with particular structure (this is essentially a form of adaptive estimation). If you want to pursue that approach you'll likely need to narrow the scope of the problem and do some kind of study (either by looking at asymptotic properties or by use of simulation) of the properties of some proposed adaptive estimator.

• Let me give some background on my question. In price index number analysis, there are a variety of formulas based on different concepts of means and different statitical agencies use one of these formulas for calculating exactly the same thing. So type of variables or purpose seems not enough to convince every one. – user41838 Mar 20 '14 at 4:11
• Besides, any of the following that we accept as the criteria: a) Different means serves different purposes b) Type of variable is important 3- Distribution of the data has a role. Why can't they be put into a formal framework? – user41838 Mar 20 '14 at 4:31