Difference between revisions of "Gini Index"

Revision as of 13:13, 15 June 2021

Definition

For the purpose of measuring concentration, the Gini Index (also Gini coefficient) is an index defined in terms of the Lorentz curve of distribution values.

Details

More precisely, if we have n values $E_i$ summing up to a total value of

E_T = \sum^{n}_{i=1} E_{i}

and the fractional value $w_i$ is defined as

w_{i} = \frac{E_i}{E_T}

Then the Gini index is defined as the area under the Lorenz curve which is geometrically reduced to

G = 1 + \frac{1}{n} \sum^{n}_{i=1} (1 - 2 i) w_{i}

Alternative Formula

Gini's absolute mean difference is defined as

\Delta = \frac{1}{n^2} \sum^{n}_{i=1} \sum^{n}_{j=1} | E_i - E_j |

Usage

None

Variations

None

Issues and Challenges

NB: Sometimes the formula appears also with the opposite sign!

Implementations

Open Source implementations of the Gini index are available in

the R package Ineq
the Python library Concentration Library

References

@@ Line 1: / Line 1: @@
 == Definition ==
-For the purpose of measuring name, sector or geographic concentration, the '''Gini Index''' (also Gini coefficient) is an index defined in terms of the Lorentz curve of the exposures.
+For the purpose of measuring concentration, the '''Gini Index''' (also Gini coefficient) is an index defined in terms of the Lorentz curve of distribution values.
 == Details ==
-More precisely, if we have n exposures <math>E_i</math> summing up to a total exposure of
+More precisely, if we have n values <math>E_i</math> summing up to a total value of
 :<math>
 E_T = \sum^{n}_{i=1} E_{i}
 </math>
-and the fractional exposures <math>w_i</math> are defined as
+and the fractional value <math>w_i</math> is defined as
 :<math>
 w_{i} = \frac{E_i}{E_T}
@@ Line 14: / Line 14: @@
 :<math>
 G =  1 + \frac{1}{n}   \sum^{n}_{i=1} (1 - 2 i) w_{i}
+</math>
+== Alternative Formula ==
+Gini's ''absolute mean difference'' is defined as
+:<math>
+\Delta =  \frac{1}{n^2}   \sum^{n}_{i=1} \sum^{n}_{j=1} | E_i - E_j |
 </math>