Statistics of Ordinal Variation

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.

Get full access to this article

View all access options for this article.

References

Agresti, A. 1990. Categorical Data Analysis. New York: John Wiley & Sons.

Agresti, A. and B. F. Agresti . 1977. “Statistical Analysis of Qualitative Variation.” Pp. 204-237 in Sociological Methodology, edited by K. F. Schuessler . San Francisco: Jossey-Bass.

Allison, Paul D. 1978. “Measures of Inequality.” American Sociological Review 43:865-880.

Atkinson, A. B. 1970. “On the Measurement of Inequality.” Journal of Economic Theory 2:244-263.

Berger, R. L. and D. D. Boos . 1994. “P Values Maximized Over a Confidence Set for the Nuisance Parameter.” Journal of the American Statistical Association 89:1012-1016.

Berry, K. J. and P. W. Mielke, Jr. 1992a. “Assessment of Variation in Ordinal Data.” Perceptual and Motor Skills 74:63-66.

Berry, K. J. and P. W. Mielke, Jr. 1992b. “Indices of Ordinal Variation.” Perceptual and Motor Skills 74:576-578.

Blair, J. and M. G. Lacy . 1996. “Measures of Variation for Ordinal Data as Functions of the Cumulative Distribution.” Perceptual and Motor Skills 82:411-418.

Dalton, H. 1920. “The Measurement of the Inequality of Incomes.” Economic Journal 30:348-361.

10.

Davis, J. A. and T. W. Smith . 1994. General Social Surveys 1972-1994 [MRDF]. Chicago: National Opinion Research Center.

11.

Greer, D. and M. G. Lacy . 1993. “The Fit Between Perceived Equity and the Objective Supply of Urban Recreation Resources.” Paper presented at the Seventh Canadian Congress on Leisure Research, May 13-15, Winnipeg, Canada.

12.

Kvålseth, T. O. 1995a. “Coefficients of Variation for Nominal and Ordinal Categorical Data.” Perceptual and Motor Skills 80:843-847.

13.

Kvålseth, T. O. 1995b. “Comment on the Coefficient of Ordinal Variation.” Perceptual and Motor Skills 81:621-622.

14.

Leik, R. K. 1966. “A Measure of Ordinal Consensus.” Pacific Sociological Review 9:85-90.

15.

Mueller, J. H. , and K. F. Schuessler . 1961. Statistical Reasoning in Sociology. Boston: Houghton Mifflin.