Differentiation between populations and its measurement

Abstract

When applied to a family of sets, the term “differentiation” designates a measure of the totality of those members which appear in only one of the sets. This basic set theoretic concept involves the formation of intersections, unions, and complements of sets. However, populations as special kinds of sets may share types, but they do not share the carriers of these types; intersections of different populations are thus always empty. The resulting conceptual dilemma is resolved by considering the joint representation of members of different populations that have the same type; populations then intersect with respect to joint representation of types. Two forms of representation reflect relative and absolute characteristics of differentiation by accounting for the distributions of types as relative frequencies within populations (as is commonly done) and as absolute frequencies (including effects of population sizes on differentiation), respectively. Corresponding classes of differentiation measures are developed, and existing measures are discussed in relation to these classes. In particular, the affinity of the measurement of distances between populations and the special case of differentiation of two-population families is examined in order to distinguish between the notions of distance and differentiation.