A Note on Distance Variance for Categorical Variables
Abstract
This study investigates the extension of distance variance, a validated spread metric for continuous and binary variables, to quantify the spread of general categorical variables. We provide both geometric and algebraic characterizations of distance variance, revealing its connections to some commonly used entropy measures, and the variance-covariance matrix of the one-hot encoded representation. However, we demonstrate that distance variance fails to satisfy the Schur-concavity axiom for categorical variables with more than two categories, leading to counterintuitive results. This limitation hinders its applicability as a universal measure of spread.
Fields of Study & Subjects
Statistical Methods and Applications
Probability and Statistical Research
Advanced Statistical Methods and Models
Publication details
- Journal / source
- Statistical Science
- Volume
- 41
- Issue
- 1
- Pages
- 211–217
- Publication date
- 14 January 2026
- Language
- EN