# Where does the Gini coefficient come from?

I understand what a ROC curve is. However, I do not understand the Gini coefficient in the context of binary classification.

All the resources I have checked state that $$Gini = 1 - (2 \times AUC_{ROC})$$.

How is this equality derived? As an economist, it troubles me to think of a Gini coefficient in this context.

• Aside: Gini, Corrado. (1912) "On the measure of concentration with especial reference to income and wealth" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955). Mar 2, 2022 at 20:54
• Watch out: the name Gini coefficient has been applied to at least three different measures. Trust equations used, not names. Mar 2, 2022 at 21:25
• @NickCox Is that right? Can you recommend a good "Hey look at these three different formal measures!"-type article or blog? Mar 2, 2022 at 21:47
• Not readily. The main point is that there are literatures that don’t really touch, notably income inequality and CART. Mar 2, 2022 at 22:17

Also, if you have fitted probabilities $$\hat p_i$$ for each individual, and you have a well-calibrated model (ie, $$\hat p_i$$ really does estimate $$P[Y=1|X=x_i]$$), there is some relationship between the two in concept. If you have a well-calibrated model, you can summarise how well it predicts by considering the variability in the predictions – since, ex hypothesi, different predictions for different people reflects genuine discriminatory power. So, you can look at the $$\hat p$$ and ask how unequal they are, with more inequality implying better discrimination.