The pp-folded cumulative distribution function and the mean absolute deviation from the pp-quantile

The aims of this short note are two-fold. First, it shows that, for a random variable XX, the area under the curve of its folded cumulative distribution function equals the mean absolute deviation (MAD) from the median. Such an equivalence implies that the MAD is the area between the cumulative distribution function (CDF) of XX and that for a degenerate distribution which takes the median as the only value. Secondly, it generalises the folded CDF to a pp-folded CDF, and derives the equivalence between the area under the curve of the pp-folded CDF and the weighted mean absolute deviation from the pp-quantile (MADp). In addition, such equivalences give the MAD and MADp simple graphical interpretations. Some other practical implications are also briefly discussed.