By definition, a categorical variable has a measurement scale that consists of a set of categories, either nominal (i.e., categories without any natural ordering) or ordinal (i.e., categories that are ordered). For a categorical variable with n categories and the probability distribution P n = (p 1, …, p n ) where p i ≥ 0 for i = 1, …, n and\( \sum\limits_{i = 1}^n {p_i = 1}, \) some measurement of variation (dispersion) is sometimes of interest. Any such measure will necessarily depend on whether the variable (or set of categories or data) is nominal or ordinal.
Nominal Case
In the nominal case, variation is generally considered to increase strictly as the probabilities (or proportions) p i (i = 1, …, n) become increasingly equal, with the variation being maximum for the uniform distribution P n 1 = (1 ∕ n, …, 1 ∕ n) and minimum for the degenerate distribution P n 0 = (0, …, 0, 1, 0, …, 0) and for any given n. In terms of majorization theory (Marshall and Olkin 1979, Ch. 1), this...
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References and Further Reading
Agresti A (2002) Categorical data analysis, 2nd edn. Wiley, Hoboken, NJ
Blair J, Lacy MG (1996) Measures of variation for ordinal data as functions of the cumulative distribution. Percept Mot Skills 82:411–418
Kvälseth TO (1995a) Coefficients of variation for nominal and ordinal categorical data. Percept Mot Skills 80:843–847
Kvälseth TO (1995b) Comment on the coefficient of ordinal variation. Percept Mot Skills 81:621–622
Kvälseth TO (1998) On difference – based summary measures. Percept Mot Skills 87:1379–1384
Leik RK (1966) A measure of ordinal consensus. Pacific Sociol Rev 9:85–90
Marshall AW, Olkin I (1979) Inequalities: theory of majorization and its applications. Academic Press, San Deigo, CA
Weisberg HF (1992) Central tendency and variability. (Sage University Paper Series No. 07-083). Sage Publications, Newbury Park, CA
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Kvålseth, T.O. (2011). Variation for Categorical Variables. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_608
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