Suppose that a population of individuals may be grouped according to some vector of characteristics into "clusters," such that each cluster is very "similar" in terms of the attributes of its members, but different clusters have members with very "dissimilar" attributes. In that case we say that the society is polarized. Our purpose is to study polarization, and to provide a theory of its measurement. Our contention is that polarization, as conceptualized here, is closely related to the generation of social tensions, to the possibilities of revolution and revolt, and to the existence of social unrest in general. We take special care to distinguish our theory from the theory of inequality measurement. We derive measures of polarization that are easily applicable to distributions of characteristics such as income and wealth.