No abstract

Whereas measures of variation in nominal data have long been recognized and used by sociologists, measures of variation for ordered categorical data have received little attention. The authors discuss the potential usefulness of ordinal dispersion statistics in sociology and define a broad class of such measures, some of which have previously been proposed in other forms. This article focuses on two statistics, termed l2 and l, which are [0 - 1] normed measures of concentration or dispersion, and illustrates their use for two purposes: measuring inequality and cultural consensus. The bias, variance, and use of these statistics in inference are discussed. The article concludes with a substantive application of these statistics and a comparison to the performance of conventional variation statistics.

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.

No abstract

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