  This study investigates the extension of distance variance, a validated
spread metric for continuous and binary variables [Edelmann et al., 2020, Ann.
Stat., 48(6)], 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.
