Measures of dispersion for categorical random variables based on penalty functions play a central role in establishing relevant measures of association between such variables. The literature concerning these measures provides little systematic treatment of such aspects of these measures as comparability, efficient estimation, and large-sample properties. This article provides a systematic and rigorous construction of dispersion measures based on penalty functions. Efficient estimation procedures and asymptotic properties of estimates are examined. Conditions from majorization theory that ensure a meaningful comparability of dispersion measures based on penalty functions are discussed. A large class of familiar dispersion measures is then given a new interpretation using these conditions.