Editing Gini coefficient (section)

== Definition ==
[[File:Economics Gini coefficient2.svg|thumb|right|upright=1.40|The Gini coefficient is equal to the area marked ''A'' divided by the total area of ''A'' and ''B'', i.e. <math>\text{Gini}=\tfrac{A}{A+B} </math>. The axes run from 0 to 1, so ''A'' and  ''B'' form a triangle of area <math>\tfrac{1}{2} </math> and  <math>\text{Gini} = 2A = 1-2B  </math>.]]

The Gini coefficient is an index for the degree of inequality in the distribution of income/wealth, used to estimate how far a country's wealth or income distribution deviates from an equal distribution.<ref>{{Cite web |title=Glossary {{!}} DataBank |url=https://databank.worldbank.org/metadataglossary/world-development-indicators/series/SI.POV.GINI |access-date=2023-04-13 |website=databank.worldbank.org}}</ref>

The Gini coefficient is usually defined [[mathematics|mathematically]] based on the [[Lorenz curve]], which plots the proportion of the total income of the population (y-axis) that is cumulatively earned by the bottom ''x'' of the population (see diagram).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Gini Coefficient |url=https://mathworld.wolfram.com/ |access-date=2023-04-13 |website=mathworld.wolfram.com |language=en}}</ref> The line at 45 degrees thus represents perfect equality of incomes. The Gini coefficient can then be thought of as the ratio of the area that lies between the line of equality and the Lorenz curve (marked ''A'' in the diagram) over the total area under the line of equality (marked ''A'' and ''B'' in the diagram); i.e., {{nowrap|G {{=}} ''A''/(''A'' + ''B'')}}. If there are no negative incomes, it is also equal to 2''A'' and {{nowrap|1 − 2''B''}} due to the fact that {{nowrap|''A'' + ''B'' {{=}} 0.5}}.<ref>{{Cite web |title=5. Measuring inequality: Lorenz curves and Gini coefficients – Working in Excel |url=https://www.core-econ.org/doing-economics/book/text/05-02.html |access-date=2023-04-26 |website=www.core-econ.org |language=en}}</ref>

Assuming non-negative income or wealth for all, the Gini coefficient's theoretical range is from 0 (total equality) to 1 (absolute inequality). This measure is often rendered as a percentage, spanning 0 to 100. However, if negative values are factored in, as in cases of debt, the Gini index could exceed 1. Typically, we presuppose a positive mean or total, precluding a Gini coefficient below zero.<ref>{{Cite web |title=cumulative distribution function - How to compute the Wealth Lorenz curve with negative values? |url=https://stats.stackexchange.com/q/235470 |access-date=2022-11-30 |website=Cross Validated |language=en}}</ref>

An alternative approach is to define the Gini coefficient as half of the [[relative mean absolute difference]], which is equivalent to the definition based on the [[Lorenz curve]].<ref>{{citation | last = Sen | first = Amartya | author-link = Amartya Sen | title = On Economic Inequality | publisher = Oxford University Press | place = Oxford | edition = 2nd | year = 1977}}</ref>  The mean absolute difference is the average [[absolute difference]] of all pairs of items of the population, and the relative mean [[absolute difference]] is the mean absolute difference divided by the [[arithmetic mean|average]], <math>\bar{x}</math>, to normalize for scale. If ''x''<sub>''i''</sub> is the wealth or income of person ''i'', and there are ''n'' persons, then the Gini coefficient ''G'' is given by:

:<math>G = \frac{\displaystyle{\sum_{i=1}^n \sum_{j=1}^n \left| x_i - x_j \right|}}{\displaystyle{2 n^2 \bar{x}}} = \frac{\displaystyle{\sum_{i=1}^n \sum_{j=1}^n \left| x_i - x_j \right|}}{\displaystyle{2 n \sum_{i=1}^n x_i}} </math>

When the income (or wealth) distribution is given as a continuous [[probability density function]] ''p''(''x''), the Gini coefficient is again half of the relative mean absolute difference:

:<math>G = \frac{1}{2\mu}\int_{-\infty}^\infty\int_{-\infty}^\infty p(x)p(y)\,|x-y|\,dx\,dy</math>

where <math>\textstyle\mu=\int_{-\infty}^\infty x p(x) \,dx</math> is the mean of the distribution, and the lower limits of integration may be replaced by zero when all incomes are positive.<ref>Dorfman, Robert. “A Formula for the Gini Coefficient.” ''The Review of Economics and Statistics'', vol. 61, no. 1, 1979, pp. 146–49. ''JSTOR'', {{doi|10.2307/1924845}}. Accessed 2 Jan. 2023.</ref>