This note constructs a simple two class example in which the Gini index is held constant while the size of the rich and poor populations change, in order to illustrate how very different societies can have the same Gini index and produce very similar estimates of standard inequality averse Social Welfare Functions. The rich/poor income ratio can vary by a factor of over 12, and the income share of the top one per cent can vary by a factor of over 16, with exactly the same Gini index. Focussing solely on the Gini index can thus obscure perceptions—e.g. of important market income trends or large changes in the redistributive impact of the tax and transfer system. Hence, analysts should supplement the use of an aggregate summary index of inequality with direct examination of the segments of the income distribution which they think are of greatest importance.