# Variation for Categorical Variables

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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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• Weisberg HF (1992) Central tendency and variability. (Sage University Paper Series No. 07-083). Sage Publications, Newbury Park, CA

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### Cite this entry

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