A two-dimensional measure of polarization:

"The link between economic distribution and social conflict—and the notion that this link arises from individuals' sense of identification with those similar to them and their feelings of alienation from individuals with different characteristics—has spurred a literature on polarization, a concept distinct from inequality. This literature, with few exceptions, has nearly exclusively focused on polarization along one (i.e., economic) dimension, despite ample evidence that identification and alienation are often formulated along noneconomic attributes. This paper extends previous work by presenting and discussing a measure of polarization that allows analysis of the distribution of society along two dimensions—an economic variable (e.g., income) and an immutable variable with social significance (e.g., skin color). The measure is discussed in light of four axioms that specify the types of distributional changes that should reasonably translate into a higher degree of socioeconomic polarization. Applying the measure to a family of functions that can represent both unimodal and bimodal population distributions, the measure satisfies the four axioms—briefly summarized as a shrinking of the middle class, greater concentration of the population around poles, greater distance between the poles, and higher correlation between the two variables—under certain parametric restrictions. Unlike the existing studies, which explore multidimensional polarization, we propose a polarization measure that treats the social attribute as continuous (and hence with ordinal properties), thus being able to capture both identification and alienation in social and economic terms. " from authors' abstract