Compelling example for the geometric mean? Does anyone have a nice example where the solution is the geometric mean? There are nice examples where the solution is the harmonic mean; see, e.g., Arithmetic mean vs Harmonic mean An example similar to that (not rooted in geometry) would be great. 
 A: Suppose you are given $n$ independent samples $X_1, \dots X_n$ from a normal distribution with known variance $\sigma^2$ but unknown mean $\mu$. Then the maximum likelihood estimate for $\mu$ is given by the arithmetic mean $\frac{X_1 + \dots + X_n}{n}$ of the samples. This is a common example of how arithmetic means are relevant to probability and statistics. 
Now suppose you are instead given $n$ independent samples from a lognormal distribution given by the exponential of a normal distribution with known variance $\sigma^2$ but unknown mean $\mu$. Then the maximum likelihood estimate for $\exp(\mu)$ is given by the geometric mean of the samples. This is essentially the same fact as the previous but exponentiated. 
In the same way that the central limit theorem leads to approximately normally distributed quantities if they are given as the sum of many small independent contributions, it also leads to approximately lognormally distributed quantities if they are given as the product of many small independent contributions. See Wikipedia for examples. 
