When dealing with quantizers, and in many other communications problems, the interest is usually on the mean-squared distortion or mean-squared error, rather than mean absolute error or anything else. Gallager explains the reason here, probably perfectly if I were to extrapolate from the quality of the rest of his lectures, but I don't get it.
He says that the mean-squared distortion of the quantizer maps nicely to the mean-squared distortion of the waveform we sample&quantize, and this mapping wouldn't be as nice if we used another metric such as mean absolute distortion. Why is that? Shouldn't the mean-absolute difference between the original and the sampled-quantized waveform also map to the mean-absolute difference of the quantizer? What makes mean-square different in this sense?