if "Average" is the term for taking the sum of many values and multiplying by the inverse-count, what is the term for multiplying many values and raising to the inverse-count power?
so instead of the process of Averaging: $$\left(\sum\limits_1^n N_i\right)\frac{1}{n}$$ I'm looking for the name of this process: $$\left( \prod\limits_1^n N_i\right)^\frac{1}{n}$$
This may have already been asked, but I've been searching for about a half hour with no luck or idea on how to word the search. This is apparently a very difficult term to search for.
What you are calling "averaging" is usually called the arithmetic mean.
The equivalent for products is called the geometric mean.
The two are famously related by the AM-GM inequality:
$$\sqrt[n]{\prod_{i=1}^n N_i}\leqslant\frac1n\sum_{i=1}^n N_i.$$
There are other means, such as the Harmonic mean $$\frac{n}{\sum_{i=1}^n\frac1{N_i}},$$ which is $\leqslant$ the geometric mean. These three together are known as the Pythagorean means. See here for a list of popular means.