Why use RMS values instead of "mean of absolute"
We use RMS when we are interested in the power delivered by a source (into an ohmic load). For a DC voltage that power is proportional to V^2: P = I * V, and I = V / R, hence V appears twice in the formula.

So the power at any one moment is proportional to V^2. Hence the average power is proportional to mean( V^2 ). When we want to know which DC voltage would cause the same power, we have the equation 

mean( DC^2 ) = mean( V^2 )


which is easily reduced to

DC = sqrt( mean( V^2 ))


in other words, the root of the mean of the square => root-mean-square.

Summary: the RMS voltage is the DC voltage that would cause the same power (in an ohmic load) as the AC voltage we are studying.