We address income inequality, particularly in heavily skewed distributions where traditional measures like the Gini index - based on the population mean - may be inadequate. We explore three median-based inequality indices that reflect three views of measuring income inequality: in the first, the median income of the poor is compared with the median income of the entire population, in the second the comparison is made with the median income of those who are not poor, in the third the comparison is with the median of the same proportion of the richest. Several properties of the indices have been derived and discussed. The indices offer many advantages, the main one is that they are applicable to population distributions with any degree of tail heaviness, even not having any finite moment. An application to capital income data from European countries demonstrate the interpretation and the efficacy of the indices.