The p-folded cumulative distribution function and the mean absolute deviation from the p-quantile
Jinghao Xue and Mike Titterington
Statistics and Probability Letters
The aims of this short note are twofold. First, it shows that, for a random variable X, the area under the curve of its folded cumulative distribution function (or the mountain plot) equals the mean absolute deviation from the median (MAD). Such an equivalence implies that the MAD is the area (or a measure of absolute difference) between the cumulative distribution function (CDF) of X and that for a degenerate distribution which takes the median as the only value. Secondly, it generalises the folded CDF to a p-folded CDF, and derives the equivalence between the area under the curve of the p-folded CDF and the (weighted) mean absolute deviation from the p-quantile (MAD_p). In addition, such equivalences give the MAD and MAD_p simple graphical interpretations. Some other practical implications are also briefly discussed.