Many standard inequality measures can be written as ratios with the mean in the denominator. When one income moves away from equality, both the numerator and the denominator may vary in the same direction and such indices may decrease. This anomalous behaviour is not shared by median-normalised inequality measures developed in this paper, where the mean at the denominator is replaced by the median. However, median-normalised inequality measures do not respect the principle of transfers. We show that the absolute Gini and the mean logarithmic deviation, or second Theil index, are the only measures that both avoid anomalous behaviour when one income is varied and also satisfy the principle of transfers. An application shows that the increase in inequality in the United States over recent decades is understated by the Gini index and that the mean logarithmic deviation index should be preferred in practice.