The Relationship between Gini's Mean Difference and the Absolute Deviation from a Quantile

The aim of this note is to investigate the relationship between Gini's Mean Difference (GMD), the mean absolute deviation (MAD), the least absolute deviation (LAD), and the absolute deviation from a given quantile (QUAD). The latter measures can all be interpreted as equivalents either to the GMD of a transformed distribution, or alternatively, to a between-group GMD (BGMD) measure, according to the particular partition of the data. As such they all possess properties of the GMD but each omits the intra-group variability – and, of course, they give rise to different regression techniques. It is argued that the loss of the intra-group information is too heavy a price to pay, and that the analyst using one of these techniques should justify the omission of the intra-group variability from the analysis.