Measures of Variation for Ordinal Data as Functions of the Cumulative Distribution

Abstract

A new measure of ordinal variation, the LSQ, is developed using a geometric representation involving the cumulative distribution function. Connections among it and previously suggested measures, the LOV, IOV, and COV, are clarified. This geometric perspective helps demonstrate that all these statistics measure the distance between the observed cumulative distribution and that corresponding to the maximally dispersed distribution, given the sample size and the number of categories for the ordinal variable. From this perspective, it is clear that none of these measures relies on supra-ordinal assumptions concerning intercategory distances. Recent questions concerning scale invariance and unreasonable values for these measures are also clarified.