It seems like taking the n-th root of the product of n values is analogous to the traditional form of averaging. For example:
$$ \frac{5+5+5}{3} = 5 = \sqrt[\leftroot{0}\uproot{0}3]{(5*5*5)} $$
But they do not always produce the same results
$$ \frac{1+2+3}{3} = 2 $$ $$ \sqrt[\leftroot{0}\uproot{0}3]{(1*2*3)} = 1.817... $$
Also, the n-th root "average" cannot handle negative numbers. Anyway, what are the benefits of this latter form of averaging versus the standard way?
(Context: I'm in a course where "perplexity" was intuitively defined as the "average amount of surprise". It made me wonder why we don't just use the standard way of averaging.)
The geometric mean can be thought of as $e^y$ where $y$ is the average of the natural logarithms of your values. That is $$\left(\prod_{i=1}^n x_i\right)^\frac 1 n = e^{ {\left( \sum_{i=1}^n \ln x_i \right)} \over n}$$Thus this sort of average is useful when your values can be thought of as powers of some base, and you want to find the average value of the exponent.