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.)
Best Answer
One possible answer is for defining unbiased estimators of probability distributions. Often times you want some transformation of the data that gets you closer to, or exactly to, a normal distribution. For example, products of lognormal variables are again lognormal, so the geometric mean is appropriate here (or equivalently, the additive mean on the natural log of the data). Similarly, there are cases where the data are naturally reciprocals or ratios of random variables, and then the harmonic mean can be used to get unbiased estimators. These show up in actuarial applications, for example.