When measuring inequality using conventional inequality measures, ethical assumptions about distributional preferences are often implicitly made. In this paper, we ask whether the ethical assumptions underlying the concentration index for income-related health inequality and the Gini index for income inequality are supported in a representative sample of the Swedish population using an internet-based survey. We find that the median subject has preferences regarding income-related health inequality that are in line with the ethical assumptions implied by the concentration index, but put higher weight on the poor than what is implied by the Gini index of income inequality. We find that women and individuals with a poorer health status put higher weight on the poor than men and healthier individuals. Ethically flexible inequality measures, such as the s-Gini index and the extended concentration index, imply that researchers have to choose from a toolbox of infinitely many inequality indices. The results of this paper are indicative of which indices (i.e. which parameter values) reflect the views of the population regarding how inequality should be defined.