If we regard taking the absolute value as a squaring followed by taking the positive square root, then basically we have "root mean square" (STD) versus "root square mean" (MAD), that is all. The one calculation takes the square root after the mean, the other moves it before.
If we extend MAD to vectors, then we average the vector norms.
What is the norm? It is the root mean square of the vector components. So then MAD is then the "root square mean" of "root mean squares".
If we extend MAD to vectors, then we average the vector norms.
What is the norm? It is the root mean square of the vector components. So then MAD is then the "root square mean" of "root mean squares".