We propose confidence sets for inequality indices and their differences, which are robust to the fact that such measures involve possibly weakly identified parameter ratios. We also document the fragility of decisions that rely on traditional interpretations of - significant or insignificant - comparisons when the tested differences can be weakly identified. Proposed methods are applied to study economic convergence across U.S. states and non-OECD countries. With reference to the growth literature which typically uses the variance of log per-capita income to measure dispersion, results confirm the importance of accounting for microfounded axioms and shed new light on enduring controversies surrounding convergence.