Average Logarithms and the Geometric Mean

I recently came across a surprising connection between the geometric mean and logarithms. At least, it was surprising to me, but isn’t that surprising once you see the derivation (a lot of things in math are this way).

The connection is this. Take a set of observations \(y_1, y_2, \dots, y_n\). Assume that these are all positive numbers. Then \[ \langle \log(y) \rangle = \log(G).\] Here the angle brackets indicate the mean (expectation value), and \(G\) is the geometric mean of the observations. In words, the average logarithm of the observations is equal to the log of the geometric mean.

Showing this just requires using the definition of the mean and basic properties of logarithms: \[ \begin{split} \langle \log(y) \rangle & = \frac{1}{n}\left( \log(y_1) + \log(y_2) + \dots + \log(y_n) \right) \\ & = \frac{1}{n} \log(y_1 y_2 \cdots y_n) \\ & = \log((y_1 y_2 \cdots y_n)^{1/n}) \\ & = \log(G) \end{split} \]

See my previous posts on the geometric mean here:

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