Gini Coefficient

Basic Concepts

The Gini Coefficient of Inequality (aka the Gini Index) is a statistic that measures the inequality within a population, based on some non-negative measurement. For a finite sample or population of size n with measurements x₁, x₂, …, x_n in ascending order, the Gini index can be defined in any one of the following equivalent ways:

We also have

or the Gini coefficient is equal to

The Gini index of a sample is a consistent, but biased, estimator of the population Gini index. A less biased version of the Gini index is given by

Jackknifing and bootstrapping can be used to obtain unbiased estimates of the population Gini index as well as confidence intervals. See below for more details.

The Gini index takes values between 0 and 1. 1 represents the maximum level of inequality, while 0 represents the minimum level of inequality.

The Gini index is commonly used to measure wealth or income inequality in a population. A Gini index for income of 0 means that all the members of the population have the same income, an index of 0 means that one member of the population gets all the income and everyone else has no income.

Related Metrics

There are a number of other metrics that are related to the Gini coefficient. In particular, the mean absolute difference (aka the absolute mean difference) is defined as

Also, we define the relative mean absolute difference as

and so, the relative mean absolute difference is equal to twice the Gini coefficient.

Related Topics

• Lorenz Curve and example
• Standard error and confidence interval

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Mirzaei, S., Borzadaran G. R. M., Aminib, M., Jabbarib, H. (2017) A comparative study of the Gini coefficient estimators based on the regression approach. Communications for Statistical Applications and Methods. Vol. 24, No. 4, 339–351.
http://www.csam.or.kr/journal/download_pdf.php?doi=10.5351/CSAM.2017.24.4.339

Buchan, I. E. (2016) Gini coefficient of inequality. Stats Direct Limited
https://www.statsdirect.com/help/nonparametric_methods/gini_coefficient.htm

Wikipedia (2020) Gini coefficient
https://en.wikipedia.org/wiki/Gini_coefficient