I admit I don't really know what type of answer your looking for. So, I'll say something that might very well be entirely irrelevant for your purposes but which I enjoy. At least, it'll provide some context for the power means you asked about.
These generalized power means are basically the discrete (finitary) analogs of the L^p norms. So, for instance, it's with these norms that you prove (using, say, elementary calculus) the finitary version of Holder's inequality, which is really important in analysis, because it leads (via a limiting argument) to the more important fact that $L^p$ and $L^q$ spaces (which are continuous analogs of these finitary $l^p$ spaces) are dual for $p,q$ conjugate exponents.
This duality is really important: one example is that if you are trying to prove something about the $L^p$ spaces that is preserved under duality, you just have to restrict yourself to the case $1 \leq p \leq 2$. The theory of singular integral operators provides examples of this: basically, it's easy to prove they are bounded (i.e., reasonably well-behaved) for $p=2$ by Fourier analysis; you prove that they're "weak-bounded" on $L^1$ (in some sense which I won't make precise); then you apply to general results on interpolation to get boundedness in the range $1-2$; finally, this duality operation gives it for $p>2$ as well.
Also, root-mean-square speed is used to define temperature in physics.