On the measurement of inequality for ordinal data

Atkinson’s Theorem (Atkinson, 1970) is a classic result in inequality measurement. It establishes Lorenz dominance as a useful criterion for judging whether one distribution is more unequal than another, because if distribution A Lorenz dominates distribution B, then all indices in a broad class of measures must agree that A is less unequal than B. Yet recent research shows that standard inequality theory cannot be used with ordinal data such as self-reported health status or educational attainment and therefore a new theory is being developed (Apouey, 2007; Naga and Yalcin, 2008). In this paper we formulate and prove an analog of Atkinson’s theorem in an ordinal framework: Allison Foster relation (Allison and Foster, 2004) is the widest relation with which all indices fulfilling median preserving spread agree, that is, these indices do not decrease whenever probability mass is transferred away from the median.