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

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.

Answer 2

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

Answer 3

Arithmetic mean is correct.

Your scale is artificial:

• It is bounded, from 0 and 10
• 8.5 is intuitively between 8 and 9

But for other scales, you would need to consider the correct mean to use.

Some other examples

In counting money, it has been argued that wealth has logarithmic utility. So the median between Bill Gates' wealth and a bum in the inner city would be a moderately successful business person. (Arithmetic average would hive you Larry Page.)

In measuring sound level, decibels already normalizes the effect. So you can take arithmetic average of decibels.

But if you are measuring volume in watts, then use quadratic means (RMS).

Answer 4

The answer depends on the context and your purpose. Percent changes were mentioned as a good time to use geometric mean. I use geometric mean when calculating antennas and frequencies since the percentage change is more important than the average or middle of the frequency range or average size of the antenna is concerned. If you have wildly varying numbers, especially if most are similar but one or two are "flyers" (far from the range of the others) the geometric mean will "smooth" the results (not let the different ones exert a change in the results more than they should). This method is used to calculate bullet group sizes (the "flyer" was probably human error, not the equipment, so the average is ""unfair" in that case). Another variation similar to geometric mean is the root mean square method. First you take the square root of the numbers, take THAT mean, and then square your answer (this provides even more smoothing). This is often used in electrical calculations and most electical meters are calculated in "RMS" (root mean square), not average readings. Hope this helps a little. Here is a web site that explains it pretty well. standardwisdom.com