The authors examine ways to measure how “segregated” is a school district. One could imagine complete segregation as being the case where each school in a district hosts a different ethnic group, and nonsegregation as the case where all schools in a district have the same ethnic distribution.
The authors propose a set of 6 axiomatic desiderata for measures of segregation, desiderata that appeal to intuitions about how a measure should be affected or not by certain changes in underlying conditions. For example, one axiomatic desideratum is called “symmetry”, which amounts to an ordering produced by a segregation measure being invariant to the renaming of the ethnic groups in question.
On these grounds, they find that a measure producing an ordering equivalent to that of the so-called Atkinson index (link) is necessary and sufficient to satisfy 5 out of the 6 desiderata, with the symmetry property being the one that is not satisfied. This strikes me as a major problem with Atkinson-type indices, as they require ad hoc decisions to combine or exclude ethnic categories in cases where districts differ in the combinations of groups that they contain.
The authors then discuss the appealing properties of orderings that are equivalent to that which is produced by the Mutual Information index. This index is an entropy (link) based measure that quantifies the reduction in uncertainty about a student’s race that comes from learning about what school she comes from; in a symmetric manner, it also equals “the reduction in uncertainty about a student’s school that comes from learning her race.” Measures that always produce an ordering equivalent to that which is produced by mutual information are necessary and sufficient for all 6 desiderata except the so-called “composition invariance” property. Composition invariance is a controversial property. It implies that the ordering imposed by the measure does not change when the size of an ethnic group in a given district is increased in a uniform manner in all schools in that district (e.g., if the number of whites increases by 10% in all schools in a district). Composition invariance runs counter to conceptualizations of segregation that emphasize “contact” between people of different ethnicities (the authors cite work by Coleman, Hoffer, and Kilgore). For this reason, I find mutual information-based measures to be especially appealing.
Clearly these measures can be applied to measuring any kind of segregation. A useful discussion.
David M. Frankel and Oscar Volij (2011) “Measuring school segregation,” Journal of Economic Theory, 146:1-38. (gated link)