"The rich are getting richer" implies that the population income distributions
are getting more right skewed and heavily tailed. For such distributions, the mean is not
the best measure of the center, but the classical indices of income inequality, including the
celebrated Gini index, are all mean-based. In view of this, Professor Gastwirth sounded an
alarm back in 2014 by suggesting to incorporate the median into the definition of the Gini
index, although noted a few shortcomings of his proposed index. In the present paper we
make a further step in the modification of classical indices and, to acknowledge the possibility
of differing viewpoints, arrive at three median-based indices of inequality. They avoid the
shortcomings of the previous indices and can be used even when populations are ultra heavily
tailed, that is, when their first moments are infinite. The new indices are illustrated both
analytically and numerically using parametric families of income distributions, and further
illustrated using capital incomes coming from 2001 and 2018 surveys of fifteen European
countries. We also discuss the performance of the indices from the perspective of income
transfers.