The relationship between the absolute deviation from a quantile and Gini’s mean difference

We investigate the relationship between Gini’s mean difference (GMD), the mean absolute deviation, the least absolute deviation and the absolute deviation from a quantile. The latter can all be interpreted as equivalents either to the GMD of a transformed distribution or to a between-group GMD measure, according to the particular partition of the data. They all possess properties of the GMD but each omits the intra-group variability—and they give rise to different regression techniques. We argue that the analyst using one of these techniques should justify the omission of the intra-group variability from the analysis. Copyright Sapienza Università di Roma 2013