This festschrift paper honors Seymour Spilerman – an early champion of probability distributions – exploring the use of probability distributions to advance knowledge in sociology, especially about inequality and related sociobehavioral phenomena. The paper begins by briefly summarizing the few necessary building blocks – the Coleman Box visualization of the sixfold classification formed by conceptualization/theory/empirics and substance/methods; personal quantitative and qualitative characteristics, including, within the quantitative set, cardinal and ordinal characteristics; inequality measures; and probability distributions. Next it turns to the titular and opening case – how probability distributions expanded the meaning of inequality from inequality between persons to inequality between subgroups, thereby undermining the case for the beneficial effects of income inequality. For while it may have been straightforward for social scientists to defend the beneficial incentive effects of inequality between persons it is a different matter entirely to defend inequality between subgroups. The paper then discusses six further applications in which probability distributions reveal new aspects of inequality and sociobehavioral phenomena, showing how inequality in ordinal characteristics differs from inequality in cardinal characteristics (for example, the Gini coefficient is constant), assessing new candidates for inequality measures (illustrating with the P90/P10 ratio and its sibling complementary quantile ratios), showing how theoretical predictions differ across different distributional families (for example, for proportions integrationist and segregationist), showing how the proportion poorest who gain from inequality reduction varies across distributional family, discerning in empirical data how people form ideas of the just job income for themselves (for example, whether they fix on a constant or a multiple or compare to everyone), and using toy distributions to highlight and contrast distributional properties – and to prepare newcomers to the knowledge quest.
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