[amelius]

You can use an increasing number of statistical moments.

https://en.wikipedia.org/wiki/Moment_(mathematics)

The arithmetic mean is one of them, which would be an argument in favor of it.

[john-h-k]

I don't think it is an argument in favour of it.

arith-mean = E[x] , the first moment of x geo-mean = exp(E[log x]) , so log geo-mean = first moment of log x

They are both equivalent in amount of information preserved, but arithmetic preserves additive structure whereas geo preserves multiplicative structure

[amelius]

I didn't want to imply that there is a loss of information. Yes, it's one level up in the hyperoperation chain, I suppose, but what I meant is that it's not a typical way of doing statistics, especially the higher moments.