How to interpret root-mean-square

Question

Could someone please describe the physical meaning of the RMS value of something? How does this interpretation relate to the average value of something?

Answer 1

Think of it as the average value of the size (magnitude) of something. For example consider the two sets of numbers $A=\{2,-2,2,-2\}$ and $B=\{1,-1,1,-1\}$. The average value is 0 for both A and B, but the typical member of A is twice as big as one in B. And the RMS values are 2 and 1 respectively.

The squaring makes all the numbers positives and prevents cancellations, but then you have the average of the squared sizes, not of the sizes themselves. That's why we take square roots in the end.

To use a non-constant "size" as an example, let $A=\{4,-2,2,-4\}$ and $B=\{2,-1,1,-2\}$. Means are zero again, but RMS values are $\sqrt 10$ and $\sqrt 2.5$ respectively which shows that numbers in A are "on average twice as big as those of A".