Root Mean Square 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?
$$\text{M}(\textbf{x})=\frac{1}{n}\sqrt{\sum_{i=1}^n x_i^2}=\frac{\|\textbf{x}\|_2}{n}$$
$$\text{M}(f)=\frac{1}{b-a}\sqrt{\int_{a}^b [f(x)]^2dx}=\frac{\|f\|_2}{b-a}$$
 A: It's under the radical because when you're doing something which is "kind of like average" such as RMS, you want the result of $n$ identical data points to be that same value. There are other reasons, but I think that has to be a big one.
A: Here's a physics exercise.  Recall from your high-school physics course that volts times amps equals watts.  That's with direct current.  Show that that also works with sinusoidal alternating current precisely if "volts" is take to mean the root-mean-square voltage as conventionally defined, i.e. the square root of the mean of the square of the voltage.
Now remember that "standard deviation" is root-mean square deviation.  When applying the central limit theorem to find the probability that the number of heads you get when you toss a coin 1600 times is between 1590 and 1630, show that everything works out neatly precisely if you use the conventional definition of root-mean-square.
A: Any sensible average, when given $n$ identical values, should result in the same value. The root mean square has this property, while your proposed definition does not.
A: Electrical engineers use RMS values when dealing with AC. The instantaneous power delivered to a resistive load is $p(t) = R \, i^2(t)$ (or the equivalent formulation with voltage), where $R$ is the resistance and $i$ the instantaneous current.
For most AC (as in industrial & domestic power) considerations, the average power over a cycle is of more interest than the instantaneous power. If one does the calculation (with $i(t) = i_\max \sin(2 \pi 60 t)$), then this is expressed as $p_\mathrm{average} = R \, i_\mathrm{rms}^2$, where $i_\mathrm{rms} = \frac{1}{\sqrt{2}} i_\max$, thus retaining the same form.
AC voltage/current is typically quoted as an RMS value, since this usually reflects the actual power or losses of interest involved. So a 110-120V domestic supply actually reflects a maximum voltage of about 156-170V.
