The geometric mean of the numbers (a[sub]1[/sub], a[sub]2[/sub]…a[sub]n[/sub]) is
GM = (a[sub]1[/sub] * a[sub]2[/sub]…*a[sub]n[/sub]) [sup]1/n[/sup]
What’s this used for? Why do we use the arithmetic mean all the time?
In general, I would say you use the geometric mean for sets of numbers that are considered as multiplicative factors.
I have also seen it used in scientific contexts for quantities, like stellar masses, that have a large dynamic range. In this case, an arithmetic mean would be biased heavily, and uselessly, toward the high end of the range. But the geometric mean of the stars’ masses gives you the typical “order of magnitude” that you’d expect if you pick a star at random.
You could use it whenever you’d use a logarithmic scale because proportions matter more than differences. Note that
GM = exp(AM(ln(a[sub]1[/sub]), ln(a[sub]2[/sub]), … ln(a[sub]n[/sub]))
where AM means arithmetic mean
The center frequency (point of zero phase shift between input and output) of an electrical band-pass filter is the geometric mean of its upper and lower half-power frequencies. Actually it is the geometric mean of any pair of upper-end and lower-end frequencies where both of them are reduced in amplitude from the center frequency amplitude by the same ratio.
This use is extremely common in behavioural sciences, too.
Geometric means show up in one of the standard proofs of the Pythagorean theorem.
Some assemblies of objects are better described by the statistics of the logs of their properties in general.
Take sand and gravel, for instance. Every time you break a bit of stone in half, the resulting bits are 10^0.3 or 2 x as massive as the original. If the number of breaks has a normal distribution, the masses will have a lognormal distribution (that is the logs of the masses will have a normal distribution).
When events in nature change something by a proportion, logged statistics are often lurking…
When you break a stone, the resulting chunks are twice as heavy? It’s a shame I can’t pull that trick on a small hunk of gold I’ve got.
Perhaps I missed something in your explanation??
In addition to the arithmetic mean and the geometric mean, there’s also the harmonic mean which is the reciprocal of the mean of the reciprocals of the numbers. As I recall, it’s useful when you are working with proportions.
If you want to read about means in great gory detail, see mean