Quantitative metrics for trait and identity distributions

Abstract

Understanding the role of demographic diversity in group settings requires effective quantitative metrics. Intersectional feminist theory has highlighted that demographic identities can intersect in complex ways, but most metrics used to study these traits are one-dimensional. In their paper "Diversity, identity, and data" (2025), Topaz et al. introduced two novel metrics that capture multiple aspects of demographic identities among group members: "intersecting diversity" and "shared identity". We present a mathematical framework to provide probabilistic interpretations for both metrics. Using these interpretations, we prove that these two measures are anti-correlated and establish tight bounds on their possible combined values, demonstrating that there is no clear "optimal" point that maximizes both metrics. We apply these metrics in three case studies on Hollywood movies, the television show "Survivor", and a random sample of North American companies in which we explore their bounds and anti-correlation as well as their relationship to group performance in these settings. By formalizing the mathematical structure for these metrics and demonstrating their empirical relevance, we provide a foundation for researchers across the social sciences, mathematics, and related fields to more precisely quantify distributions of intersecting traits within groups and better understand their implications for group dynamics and performance.