What do all the distributions that have the same Gini index have in common? According to the Wikipedia article about Income inequality metrics, Gini index have the next disadvantage:

As a disadvantage, the Gini index only maps a number to the properties of a diagram, but the diagram itself is not based on any model of a distribution process. The &quot;meaning&quot; of the Gini index only can be understood empirically. Additionally the Gini does not capture where in the distribution the inequality occurs. As a result, two very different distributions of income can have the same Gini index.

So, I want to ask - what do all the distributions that have the same Gini index have in common? (Empirically, they produce the same area under the Lorentz curve, but what does that mean?)
Thanks!
This is a Lorenz curve.



The Gini coefficient is the area between 'equal wealth' and your chosen line, divided by all the area under the equal wealth curve. There are other illustrations online, for sure, which would explain this.

The Wikipedia article is telling you that the actual distribution of income is not fully captured by the Gini coefficient. The same Gini coefficient could have different distributions of income. For example, it could move equally in the lower region at first and then flatten between 60 and 80 per cent of people. Or it could be that poorer people get nothing and the middle and upper classes are hoovering up the income. 

Both examples, if you set up the toy numbers properly, imply very different distributions for poor people, but produce the same Gini coefficient.