Which average to use? (RMS vs. AM vs. GM vs. HM) The generalized mean (power mean) with exponent $p$ of $n$ numbers $x_1, x_2, \ldots, x_n$ is defined as
$$ \bar x = \left(\frac{1}{n} \sum x_i^p\right)^{1/p}. $$
This is equivalent to the harmonic mean, arithmetic mean, and root mean square for $p = -1$, $p = 1$, and $p = 2$, respectively. Also its limit at $p = 0$ is equal to the geometric mean.
When should the different means be used? I know harmonic mean is useful when averaging speeds and the plain arithmetic mean is certainly used most often, but I've never seen any uses explained for the geometric mean or root mean square. (Although standard deviation is the root mean square of the deviations from the arithmetic mean for a list of numbers.)

An important special case of the AM-GM inequality is that the product of two (positive) numbers with a constant sum is at a maximum when they are equal. This comes up a lot.
