In dealing with audio signals processing and talking with some engineers about general signals processing I've encountered the Root Mean Square which is some sort of mean. Here's a summary of the RMS of discrete data points and of a function on an interval:
$\text{RMS}(\textbf{x})=\sqrt{\frac{1}{n}\sum_{i=1}^n x_i^2}=\frac{\|\textbf{x}\|_2}{\sqrt{n}}$
$\text{RMS}(f)=\sqrt{\frac{1}{b-a}\int_{a}^b [f(x)]^2dx}=\frac{\|f\|_2}{\sqrt{b-a}}$
Geometrically this makes sense to me because it's essentially the euclidean norm. What I'm wondering is why is the "averaging term" under the radical? It seems to me that the squaring then the square root put the data in the same scale as the original function, wouldn't it make more sense to define it the following way?
$\text{M}(\textbf{x})=\frac{1}{n}\sqrt{\sum_{i=1}^n x_i^2}=\frac{\|\textbf{x}\|_2}{n}$
$\text{M}(f)=\frac{1}{b-a}\sqrt{\int_{a}^b [f(x)]^2dx}=\frac{\|f\|_2}{b-a}$