Should one take the log of the mean, or the mean of the logs?

[Jesse Wursten]

Imagine you have employment numbers for a bunch of firms in a host of sectors which you want to aggregate to sector level data. Should you take logs first and then average those, or first take the average and then log that? (Assuming you want to work in logs) What exactly happens in both cases?

[Nick Cox]

Why not just say that

exp(mean(log())

gives the geometric mean, which works well, is a number on the original scale, and is often a natural parameter (e.g. for lognormals)? .

Geometry aside, the idea goes back at least to Galileo who was asked for advice on the price of a horse. One guy says 10 and another says 1000, so what is the right compromise? Galileo said 100. I may be mis-remembering the numbers, and I've forgotten the unit of measurement, presumably Euros, but it's in a letter of his somewhere.

A hat-tip to Italian Stata friends who do so much on Statalist! Carlo Lazzaro

[Carlo Lazzaro]

Dear Nick:
thanks, very flattering and unearned.
I will skim through my philosophy textbooks and try to spot the Galileo's letter you mentioned.
Thanks for being such a continuous source of learning.

[Anders Alexandersson]

I did a quick Google search: Supposedly the reference of the letter is Galileo (1627), and the unit of measurement is Italian scudo (plural: scudi).

Galilei, Galileo (1627). Lettera (intorno la stima di un cavallo). Le opere di Galileo Galilei. Prima edizione completa. Societa editrice fiorentina. Firenze, 1855.

Some details about Galileo's letter in English are in Walsh (1921) and in Mozgunov et al. (2017).
For Walsh, see the preview at https://www.amazon.com/Problem-Estim.../dp/1507503504
and for Mozgunov et al. see pages 2-3 at https://arxiv.org/pdf/1706.02104.pdf

The geometric mean is part of the Pythagorean means, and therefore originates from much earlier and murkier Greek philosophy.

[Carlo Lazzaro]

For those skilled in Italian (like Anders, I would say) the letter to which Nick refers to can be read (with a bit of effort, as it is written in an ancient Italian) at: https://books.google.it/books?id=U95...avallo&f=false

For those interested in bets and related odds (all in all, if things are fair, it still statistics!), a famous racehorse has been given the name of my distinguished compatriot: https://en.wikipedia.org/wiki/Galileo_(horse)

[Nick Cox]

Thanks very much to Anders and Carlo. The idea of geometric mean certainly arose in pure mathematics at least as far back as Greek mathematics. My suggestion here -- open to refutation, which would be welcome -- is that Galileo was the first person to use the idea in summarizing data, i.e. empirically.