We develop the measurement theory of polarization for the case in which income distributions can be described using density functions. The main theorem uniquely characterizes a class of polarization measures that fits into what we call the “identity- alienation” framework, and simultanously satisfies a set of axioms. Second, we provide sample estimators of population polarization indices that can be used to compare po- larization across time or entities. Distribution-free statistical inference results are also used in order to ensure that the orderings of polarization across entities are not simply due to sampling noise. An illustration of the use of these tools using data from 21 countries shows that polarization and inequality orderings can often differ in practice.
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