Gini Index

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

$E_{T}=\sum _{i=1}^{n}E_{i}$

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

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

Alternative Formula

Gini's absolute mean difference is defined as

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

The relative mean difference is defined as $\Delta /\mu$ where $\mu =E_{T}/n$

The Gini index is equivalently given by

$G={\frac {\Delta }{2\mu }}$

Usage

None

Variations

None

Issues and Challenges

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

Implementation

Open Source implementations of the Gini index are available in

• the R package Ineq
• the Python library Concentration Library

See Also

• Hall-Tideman Index

References

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