# Deducing an approximation to the Gini coefficient

Assuming we do not know the Lorenz curve function,

If $(X_k, Y_k)$ are the known points on the Lorenz curve, with the $X_k$ indexed in increasing order $(X_{k – 1} < X_k)$, so that:

• $X_k$ is the cumulated proportion of the population variable, for $k = 0,...,n$, with $X_0 = 0, X_n = 1$.
• $Y_k$ is the cumulated proportion of the income variable, for $k = 0,...,n$, with $Y_0 = 0, Y_n = 1$.
• $Y_k$ should be indexed in non-decreasing order $(Y_k \geq Y_{k – 1})$

If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

(taken from wikipedia)

I've also found this other formula to calculate the Gini coefficient:

$G=1-\frac{\sum^{n-1}_{k=1}Y_k}{\sum^{n-1}_{k=1}X_k}$

My question is how can I prove that both are equal? I haven't been able to deduce the latter from the former...

• I don't think they are the same. Have you tried it on some data? – Glen_b Mar 24 '15 at 1:51
• @Glen_b you're right. I've just tried with some data, and I get different results. For the first I get approx. 0.1975, and for the second one I get approx. 0.1538... – An old man in the sea. Mar 24 '15 at 11:25
• @Glen_b But is the second one still correct? – An old man in the sea. Mar 24 '15 at 11:34
• Well, if they aren't the same and the sample Gini coefficient is one thing rather than two different things, I doubt that both can be correct. But perhaps they're two different estimators of one thing. Perhaps context from where you got these could be important. – Glen_b Mar 24 '15 at 12:29
• @Glen_b From a book and some slides in a language other than english. Both didn't give a reason for the formula... – An old man in the sea. Mar 24 '15 at 14:07