Taken from Volume 1, Chapter 6, Section 3 of the Feynman Lectures on Physics.
Feynman says that in describing random, equally-probable-backwards-or-forwards motion, that,
We might therefore ask what is his average distance travelled in absolute value, that is, what is the average of $|D|$. It is, howevermore convenient to deal with another measure of "progress", the square of the distance: $D^2$ is positive for either positive or negative motion, and is therefore a reasonable measure of such random wandering.
We can show that the expected value of $D^2_N$ is just $N$, the number of steps taken.....
He seems to provide a reason that applies just the same to using absolute value. From there, beginning with that next paragraph, he doesn't make any sense at all, so I think this is central to his point. Why does he use squares of the distances instead of absolute value? And how does $D^2_N = N$?