This paper proposes a new framework for measuring income inequality. The framework is based on the unequally distributed (UD) incomes that are obtained by removing the equally distributed parts from incomes. We then derive the normalized norm indexes from the cumulative distribution function and the unscaled Lorenz curve of the UD incomes. The relation between the normalized norm indexes and the popular Gini coefficient and coefficient of variation (CV) shows that the Gini coefficient and CV represent only parts of income inequality. We analyze example income distributions and the Luxembourg Income Study datasets to show that the normalized norm indexes evaluate income inequality appropriately and solve the negative income problem. Keywords Coefficient of variation • Gini coefficient • Income inequality • 1 Norm • 2 Norm • Normalized norm index • Rawlsian index • Unequally distributed income JEL Classification C43 • D31 • D63 B Joongyang Park