View source for Gini coefficient

{{Short description|Measure of inequality of a distribution}}
{{distinguish|text=[[Decision tree learning#Gini impurity|Gini impurity]]}}
{{For|the list of countries sorted by the Gini coefficient|List of countries by income inequality}}
{{Use dmy dates|date=August 2017}}

[[File:Global map of high inequality countries, 2022.png|alt=|thumb|400px|World map of Gini coefficients (as a %), 2022, according to the Poverty and Inequality Platform (PIP)<ref>{{cite web |author=Haddad, Cameron Nadim; Mahler, Daniel Gerszon; Diaz-Bonilla, Carolina; Hill, Ruth; Lakner, Christoph; Ibarra, Gabriel Lara |title=Inside the World Bank’s new inequality indicator: The number of countries with high inequality |publisher=World Bank |url=https://blogs.worldbank.org/en/opendata/inside-the-world-bank-s-new-inequality-indicator--the-number-of- |date=2024-06-17 |access-date=2024-10-25}}</ref> 
{{legend-col|{{Legend|#2d6da6|<30}}|{{Legend|#91c4d9|30-35}}|{{Legend|#f2ddb6|35-40}}|{{Legend|#f2a488|40-45}}|{{Legend|#d66552|45-50}}|{{Legend|#a6243c|50+}}}}]]

{{Economics sidebar}}

In [[economics]], the '''Gini coefficient''' ({{IPAc-en|ˈ|dʒ|iː|n|i}} {{Respell|JEE|nee}}), also known as the '''Gini index''' or '''Gini ratio''', is a [[measure of statistical dispersion]] intended to represent the [[income distribution|income inequality]], the [[wealth distribution|wealth inequality]], or the [[consumption inequality]]<ref>{{cite web | url=https://databank.worldbank.org/metadataglossary/gender-statistics/series/SI.POV.GINI | title=Glossary &#124; DataBank }}</ref> within a nation or a [[social group]]. It was developed by Italian [[statistics|statistician]] and [[Sociology|sociologist]] [[Corrado Gini]].

The Gini coefficient measures the [[economic inequality|inequality]] among the values of a [[frequency distribution]], such as [[income]] levels. A Gini coefficient of 0 reflects perfect equality, where all income or wealth values are the same. In contrast, a Gini coefficient of 1 (or 100%) reflects maximal inequality among values, where a single individual has all the income while all others have none.<ref name="US Census Bureau">{{cite web|title=Current Population Survey (CPS) – Definitions and Explanations|url=https://www.census.gov/population/www/cps/cpsdef.html|publisher=US Census Bureau}}</ref><ref>Note: The Gini coefficient could be near one only in a large population where a few persons have all the income. In the special case of just two people, where one has no income, and the other has all the income, the Gini coefficient is 0.5. For five people, where four have no income, and the fifth has all the income, the Gini coefficient is 0.8. See: [https://openknowledge.fao.org/handle/20.500.14283/am352e Bellù, L.G. and Liberati, P. 2006. ''Inequality Analysis: The Gini Index''. EASYPol: Resources for policy making. Rome, FAO.]</ref>

Corrado Gini proposed the Gini coefficient as a measure of [[social inequality|inequality]] of [[income inequality metrics|income]] or [[Wealth concentration|wealth]].<ref>Gini, Corrado (1936). "On the Measure of Concentration with Special Reference to Income and Statistics", Colorado College Publication, General Series No. 208, 73–79.</ref> For [[Organisation for Economic Co-operation and Development|OECD countries]] in the late 20th century, considering the effect of [[tax]]es and [[transfer payments]], the income Gini coefficient ranged between 0.24 and 0.49, with [[Slovakia]] being the lowest and [[Mexico]] the highest.<ref name=OECD1>{{cite web|title=Income distribution – Inequality: Income distribution – Inequality – Country tables |publisher=OECD |year=2012 |url=http://stats.oecd.org/Index.aspx?QueryId=26068 |url-status=dead |archive-url=https://web.archive.org/web/20141109193609/http://stats.oecd.org/Index.aspx?QueryId=26068 |archive-date=9 November 2014 }}</ref> African countries had the highest pre-tax Gini coefficients in 2008–2009, with [[South Africa]] having the world's highest, estimated to be 0.63 to 0.7.<ref>{{cite web |year=2013 |title=South Africa Snapshot, Q4 2013 |url=http://www.kpmg.com/Africa/en/KPMG-in-Africa/Documents/2013%20Q4%20snapshots/KPMG_South%20Africa%202013Q4.pdf |url-status=dead |archive-url=https://web.archive.org/web/20160402034218/http://www.kpmg.com/Africa/en/KPMG-in-Africa/Documents/2013%20Q4%20snapshots/KPMG_South%20Africa%202013Q4.pdf |archive-date=2016-04-02 |publisher=KPMG |df=dmy-all}}{{cbignore|bot=medic}}</ref><ref>{{cite web|title=Gini Coefficient|publisher=United Nations Development Program|date=2012|url=https://data.undp.org/dataset/Income-Gini-coefficient/36ku-rvrj|url-status=dead|archive-url=https://web.archive.org/web/20140712032137/https://data.undp.org/dataset/Income-Gini-coefficient/36ku-rvrj|archive-date=12 July 2014|df=dmy-all}}</ref> However, this figure drops to 0.52 after social assistance is taken into account and drops again to 0.47 after taxation.<ref name="moneywebSA">{{cite web | url=http://www.moneyweb.co.za/moneyweb-economic-trends/the-gini-is-still-in-the-bottle | title=The Gini is still in the bottle | publisher=Money Web | date=16 July 2014 | access-date=24 November 2014 | last=Schüssler | first=Mike}}</ref> Slovakia has the lowest Gini coefficient, with a Gini coefficient of 0.232.<ref>{{Cite web |title=World Bank Open Data |url=https://data.worldbank.org/ |access-date=2023-05-09 |website=World Bank Open Data}}</ref> Various sources have estimated the Gini coefficient of the global income in 2005 to be between 0.61 and 0.68.<ref name=fao2009>{{cite web|title=Poverty, Growth, and Inequality over the Next 50 Years|first=Evan|last=Hillebrand|publisher=FAO, United Nations – Economic and Social Development Department|date=June 2009|url=ftp://ftp.fao.org/docrep/fao/012/ak968e/ak968e00.pdf|archive-url=https://web.archive.org/web/20171020065423/ftp://ftp.fao.org/docrep/fao/012/ak968e/ak968e00.pdf|url-status=dead|archive-date=2017-10-20|df=dmy-all}}</ref><ref name=undp10 />

There are multiple issues in interpreting a Gini coefficient, as the same value may result from many different distribution curves. The demographic structure should be taken into account to mitigate this. Countries with an aging population or those with an increased birth rate experience an increasing pre-tax Gini coefficient even if real income distribution for working adults remains constant. Many scholars have devised over a dozen variants of the Gini coefficient.<ref>{{cite journal|title=More than a Dozen Alternative Ways of Spelling Gini|first=Shlomo|last=Yitzhaki|journal=Economic Inequality|volume=8|year=1998|pages=13–30|url=http://siteresources.worldbank.org/INTDECINEQ/Resources/morethan2002.pdf |archive-url=https://web.archive.org/web/20120803160610/http://siteresources.worldbank.org/INTDECINEQ/Resources/morethan2002.pdf |archive-date=2012-08-03 |url-status=live}}</ref><ref>{{cite web|title=Population Aging, Mobility of Quarterly Incomes, and Annual Income Inequality: Theoretical Discussion and Empirical Findings|url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=f1a2990b709b7be14bbe46fb7fd38dfc0a780b02|publisher=Korea Institute of Public Finance|first=Myung Jae|last=Sung|date=August 2010|citeseerx=10.1.1.365.4156}}</ref><ref name=blomq81 />

==History==
The Italian statistician [[Corrado Gini]] developed the Gini coefficient and published it in his 1912 paper ''Variabilità e mutabilità'' ({{langx|en|variability and mutability}}).<ref>Gini, C. (1909). "Concentration and dependency ratios" (in Italian). English translation in ''Rivista di Politica Economica'', '''87''' (1997), 769–789.</ref><ref>{{Cite book |last=Gini |first=C |title=Variabilità e Mutuabilità. Contributo allo Studio delle Distribuzioni e delle Relazioni Statistiche |publisher=C. Cuppini |year=1912 |location=Bologna}}</ref> Building on the work of American economist [[Max O. Lorenz|Max Lorenz]], Gini proposed using the difference between the hypothetical straight line depicting perfect equality and the actual line depicting people's incomes as a measure of inequality.<ref>{{cite web |date=12 March 2015 |title=Who, What, Why: What is the Gini coefficient? |url=https://www.bbc.com/news/blogs-magazine-monitor-31847943 |access-date=30 March 2022 |website=BBC News}}</ref> In this paper, he introduced the concept of simple mean difference as a measure of variability.

He then applied the simple mean difference of observed variables to income and wealth inequality in his work ''On the measurement of concentration and variability of characters'' in 1914. Here, he presented the concentration [[ratio]], which further developed into today's Gini coefficient. Secondly, Gini observed that improving methods introduced by Lorenz, Chatelain, or Séailles could also achieve his proposed ratio.

In 1915, [[Gaetano Pietra]] introduced a geometrical interpretation between Gini's proposed ratio and between the observed area of concentration and maximum concentration. This altered version of the Gini coefficient became the most commonly used inequality index in upcoming years.<ref>{{Cite web |last=Pellegrino |first=Simone |date=2020 |title=THE GINI COEFFICIENT: ITS ORIGINS |url=https://www.bemservizi.unito.it/repec/tur/wpapnw/m70.pdf}}</ref>

According to data from the [[OECD]], the Gini coefficient was first officially used country-wide in [[Canada]] in the 1970s. Canadian index of income inequality ranged from 0.303 to 0.284 from 1976 to the end of the 1980s. The OECD has published more data on countries since the start of the 21st century. The Central European countries of [[Slovenia]], [[Czech Republic|Czechia]], and [[Slovakia]] have had the lowest inequality index of all OECD countries ever since the 2000s. [[Scandinavia]]n countries also frequently appeared at the top of the equality list in recent decades.<ref>{{Cite web |title=Inequality - Income inequality - OECD Data |url=http://data.oecd.org/inequality/income-inequality.htm |access-date=2024-04-28 |website=theOECD |language=en}}</ref>

== 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>

== Calculation ==
[[File:Gini coefficient for distribution with only two income or wealth levels.svg|thumb|upright=1.25|Richest ''u'' of population (red) equally share ''f'' of all income or wealth; others (green) equally share remainder: {{nowrap|''G'' {{=}} ''f'' − ''u''}}. A smooth distribution (blue) with the same ''u'' and ''f'' always has {{nowrap|''G'' &gt; ''f'' − ''u''}}.|right]]

While the income distribution of any particular country [[All models are wrong|will not correspond perfectly to the theoretical models]],  these models can provide a qualitative explanation of the income distribution in a nation given the Gini coefficient.

=== Example: Two levels of income ===
The extreme cases are represented by the most equal possible society in which every person receives the same income ({{nowrap|''G'' {{=}} 0}}), and the most unequal society (with ''N'' individuals) where a single person receives 100% of the total income and the remaining {{nowrap|''N'' − 1}} people receive none ({{nowrap|''G'' {{=}} 1 − 1/''N''}}).

A simple case assumes just two levels of income, low and high. If the high income group is a proportion ''u'' of the population and earns a proportion ''f'' of all income, then the Gini coefficient is {{nowrap|''f'' − ''u''}}. A more graded distribution with these same values ''u'' and ''f'' will always have a higher Gini coefficient than {{nowrap|''f'' − ''u''}}.

For example, if the wealthiest ''u ='' 20% of the population has ''f ='' 80% of all income (see [[Pareto principle]]), the income Gini coefficient is at least 60%. In another example,<ref>{{cite news |last1=Treanor |first1=Jill |date=2015-10-13 |title=Half of world's wealth now in hands of 1% of population |newspaper=The Guardian |url=https://www.theguardian.com/money/2015/oct/13/half-world-wealth-in-hands-population-inequality-report}}</ref> if ''u ='' 1% of the world's population owns  ''f ='' 50% of all wealth, the wealth Gini coefficient is at least 49%.

=== Alternative expressions ===

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the [[Lorenz curve]]. For example, (taking ''y'' to indicate the income or wealth of a person or household):
* For a population of ''n'' individuals with values <math>y_1 \leq y_2\leq \cdots \leq y_n </math>,<ref name="Wolfram Mathworld">{{cite web |title=Gini Coefficient |url=http://mathworld.wolfram.com/GiniCoefficient.html |publisher=Wolfram Mathworld}}</ref> 
::<math>G = \frac{1}{n}\left ( n+1 - 2 \left ( \frac{\sum_{i=1}^n (n+1-i)y_i}{\sum_{i=1}^n y_i} \right ) \right ). </math>
:This may be simplified to:
::<math>G = \frac{2 \sum_{i=1}^n i y_i}{n \sum_{i=1}^n y_i} -\frac{n+1}{n}.</math>

The Gini coefficient can also be considered as half the [[relative mean absolute difference]]. For a random sample ''S'' with values <math>y_1 \leq y_2\leq \cdots \leq y_n </math>, the sample Gini coefficient
:<math>G(S) = \frac{1}{n-1}\left (n+1 - 2 \left ( \frac{\sum_{i=1}^n (n+1-i)y_i}{\sum_{i=1}^n y_i}\right ) \right )</math>

is a [[consistent estimator]] of the population Gini coefficient, but is not in general [[estimator#Point estimators|unbiased]]. In simplified form:

:<math>G(S) = 1 - \frac{2}{n-1}\left ( n - \frac{\sum_{i=1}^n iy_i}{\sum_{i=1}^n y_i}\right ). </math>

There does not exist a sample statistic that is always an unbiased estimator of the population Gini coefficient.

=== Discrete probability distribution ===

For a [[discrete probability distribution]] with probability mass function <math>f ( y_i ),</math> <math qid=Q120636410>i = 1,\ldots, n</math>, where  <math>f ( y_i )</math> is the fraction of the population with income or wealth <math>y_i >0 </math>, the Gini coefficient is:
:<math>G = \frac{1}{2\mu}  \sum\limits_{i=1}^n  \sum\limits_{j=1}^n \, f(y_i) f(y_j)|y_i-y_j|</math>
where
:<math>\mu=\sum\limits_{i=1}^n y_i f(y_i).</math>
If the points with non-zero probabilities are indexed in increasing order <math>(y_i < y_{i+1})</math>, then:
:<math>G = 1 - \frac{\sum_{i=1}^n f(y_i)(S_{i-1}+S_i)}{S_n}</math>
where
:<math>S_i = \sum_{j=1}^i f(y_j)\,y_j\,</math> and <math>S_0 = 0.</math> These formulas are also applicable in the limit, as <math>n\rightarrow\infty.</math>

=== Continuous probability distribution ===

When the population is large, the income distribution may be represented by a continuous [[probability density function]] ''f''(''x'') where ''f''(''x'') ''dx'' is the fraction of the population with wealth or income in the interval ''dx'' about ''x''.  If ''F''(''x'') is the [[cumulative distribution function]] for ''f''(''x''):

:<math>F(x)=\int_0^x f(x)\,dx</math>

and ''L''(''x'') is the Lorenz function:

:<math>L(x)=\frac{\int_0^x x\,f(x)\,dx}{\int_0^\infty x\,f(x)\,dx}</math>

then the [[Lorenz curve]] ''L''(''F'') may then be represented as a function parametric in ''L''(''x'') and ''F''(''x'') and the value of ''B'' can be found by [[integral|integration]]:

:<math>B = \int_0^1 L(F) \,dF. </math>

The Gini coefficient can also be calculated directly from the [[cumulative distribution function]] of the distribution ''F''(''y''). Defining μ as the mean of the distribution, and specifying that ''F''(''y'') is zero for all negative values, the Gini coefficient is given by:
:<math>G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2 \,dy = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y)) \,dy</math>
The latter result comes from [[integration by parts]]. ''(Note that this formula can be applied when there are negative values if the integration is taken from minus infinity to plus infinity.)''

The Gini coefficient may be expressed in terms of the [[quantile function]] ''Q''(''F'') ''(inverse of the cumulative distribution function: Q(F(x)) = x)''
: <math>G=\frac{1}{2 \mu}\int_0^1 \int_0^1 |Q(F_1)-Q(F_2)|\,dF_1\,dF_2 .</math>

Since the Gini coefficient is [[income inequality metrics|independent of scale]], if the distribution function can be expressed in the form ''f(x,&phi;,a,b,c...)'' where ''&phi;'' is a scale factor and ''a, b, c...'' are dimensionless parameters, then the Gini coefficient will be a function only of ''a, b, c...''.<ref name="McDonald1974">{{cite journal |last1=McDonald |first1=James B |last2=Jensen |first2=Bartell C. |date=December 1979 |title=An Analysis of Some Properties of Alternative Measures of Income Inequality Based on the Gamma Distribution Function |url= |journal=Journal of the American Statistical Association |volume=74 |issue=368 |pages=856–860 |doi= 10.1080/01621459.1979.10481042|access-date=}}</ref> For example, for the [[exponential distribution]], which is a function of only ''x'' and a scale parameter, the Gini coefficient is a constant, equal to 1/2.

For some functional forms, the Gini index can be calculated explicitly. For example, if ''y'' follows a [[log-normal distribution]] with the standard deviation of logs equal to <math>\sigma</math>, then <math>G = \operatorname{erf}\left(\frac{\sigma }{2 }\right)</math> where <math>\operatorname{erf}</math> is the [[error function]] ( since <math> G=2 \Phi \left(\frac{\sigma }{\sqrt{2}}\right)-1</math>, where  <math>\Phi</math> is the cumulative distribution function of a standard normal distribution).<ref name='LNdist'>Crow, E. L., & Shimizu, K. (Eds.). (1988). Lognormal distributions: Theory and applications (Vol. 88). New York: M. Dekker, page 11.</ref>  In the table below, some examples for probability density functions with support on <math>[0,\infty)</math> are shown. The Dirac delta distribution represents the case where everyone has the same wealth (or income); it implies no variations between incomes.{{fact|date=January 2025}}

:{| class="wikitable" style="float: left; margin-left: 1em;"
|-
! Income Distribution function !! PDF(x) !! Gini Coefficient
|-
| [[Dirac delta function]] || <math>\delta(x-x_0),\, x_0>0</math> || 0
|-
| [[Uniform distribution (continuous)|Uniform distribution]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Uniform Distribution |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\begin{cases}
\frac{1}{b-a} & a\le x\le b \\ 0 & \mathrm{otherwise}
\end{cases}</math>
|| <math>\frac{(b-a)}{3(b+a)}</math>
|-
| [[Exponential distribution]]<ref>{{Cite web |title=Exponential Distribution {{!}} Definition {{!}} Memoryless Random Variable |url=https://www.probabilitycourse.com/chapter4/4_2_2_exponential.php |access-date=2022-11-30 |website=www.probabilitycourse.com}}</ref>
||<math>\lambda e^{-x\lambda},\,\,x>0</math>
||<math>1/2</math>
|-
| [[Log-normal distribution]]<ref name='LNdist'/><ref>For the log-normal with <math>\sigma</math> = 0, <math>\textrm{erf}(0)</math> = 0;  <math>2 \Phi(0)-1 = 2(0.5)-1</math> = 0.</ref>
||<math>\frac{1}{x\sigma\sqrt{2\pi}} e^{-\frac 12 \left(\frac{\ln\,(x)-\mu}{\sigma}\right)^2}</math>
||<math>\textrm{erf}(\sigma/2)=2 \Phi \left(\frac{\sigma }{\sqrt{2}}\right)-1</math>
|-
| [[Pareto distribution]]<ref name="mathworld.wolfram.com">{{Cite web |title=Wolfram MathWorld: The Web's Most Extensive Mathematics Resource |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\begin{cases}
\frac{\alpha k^\alpha}{x^{\alpha+1}} & x\ge k\\0 & x < k
\end{cases}</math>
||<math>\begin{cases}
1 & 0<\alpha < 1\\ \frac{1}{2\alpha -1} & \alpha \ge 1
\end{cases}</math>
|-
| [[Chi distribution]]<ref name="mathworld.wolfram.com"/>
||<math>f(x;k) = \begin{cases}
\dfrac{x^{k-1}e^{-x^2/2}}{2^{k/2-1}\Gamma\left(\frac{k}{2}\right)}, & x\geq 0 \\ 0, & x<0
\end{cases}
</math>
||<math>(-1)^k \left| I_{-1}(k,\tfrac{1}{2})\right|</math>
|-
| [[Chi-squared distribution]]<ref>{{Cite web |title=Chi-Squared Distribution -- from Wolfram MathWorld |url=https://mathworld.wolfram.com/Chi-SquaredDistribution.html |access-date=2023-01-11 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\frac{2^{-k/2} e^{-x/2} x^{k/2 - 1}}{\Gamma(k/2)}</math>
||<math>\frac{2\,\Gamma\left(\frac{1+k}{2}\right)}{k\,\Gamma(k/2)\sqrt{\pi}}</math>
|-
| [[Gamma distribution]]<ref name=McDonald1974/>
||<math>\frac{e^{-x/\theta}x^{k-1}\theta^{-k}}{\Gamma(k)}</math>
||<math>\frac{\Gamma\left(\frac{2k+1}{2}\right)}{k\,\Gamma(k)\sqrt{\pi}}</math>
|-
| [[Weibull distribution]]<ref>{{Cite web |title=Weibull Distribution: Characteristics of the Weibull Distribution |url=https://www.weibull.com/hotwire/issue14/relbasics14.htm |access-date=2022-11-30 |website=www.weibull.com}}</ref>
||<math>\frac {k} {\lambda}\, \left(\frac {x}{\lambda} \right)^{k-1} e^{-(x/\lambda)^k}</math>
||<math>1-2^{-1/k}</math>
|-
| [[Beta distribution]]<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Beta Distribution |url=https://mathworld.wolfram.com/ |access-date=2022-11-30 |website=mathworld.wolfram.com |language=en}}</ref>
||<math>\frac {x^{\alpha-1}(1-x)^{\beta-1}} {B(\alpha,\beta)}</math>
||<math>\left(\frac{2}{\alpha}\right)\frac{B(\alpha+\beta,\alpha+\beta)}{B(\alpha,\alpha)B(\beta,\beta)}</math>
|-
|[[Log-logistic distribution]]<ref>{{Cite web |title=The Log-Logistic Distribution |url=https://www.randomservices.org/random/special/LogLogistic.html |access-date=2022-11-30 |website=www.randomservices.org}}</ref>
|<math>\frac{ (\beta/\alpha)(x/\alpha)^{\beta-1} }
                       { \left (1+(x/\alpha)^{\beta} \right)^2  }</math>
|<math>1/\beta</math>
|}
{{clear}}

* <math>\Gamma(\,)</math> is the [[Gamma function]]
* <math>B(\,)</math> is the [[Beta function]]
* <math>I_k(\,)</math> is the [[Beta function|Regularized incomplete beta function]]

=== Other approaches ===

Sometimes the entire Lorenz curve is not known, and only values at certain intervals are given. In that case, the Gini coefficient can be approximated using various techniques for [[interpolation|interpolating]] the missing values of the Lorenz curve. If (''X''<sub>''k''</sub>, ''Y''<sub>''k''</sub>) are the known points on the Lorenz curve, with the ''X''<sub>''k''</sub> indexed in increasing order (''X''<sub>''k'' – 1</sub> &lt; ''X''<sub>''k''</sub>), so that:
* ''X''<sub>''k''</sub> is the cumulated proportion of the population variable, for ''k'' = 0,...,''n'', with ''X''<sub>0</sub> = 0, ''X''<sub>''n''</sub> = 1.
* ''Y''<sub>''k''</sub> is the cumulated proportion of the income variable, for ''k'' = 0,...,''n'', with ''Y''<sub>0</sub> = 0, ''Y''<sub>''n''</sub> = 1.
* ''Y''<sub>''k''</sub> should be indexed in non-decreasing order (''Y''<sub>''k''</sub> > ''Y''<sub>''k'' – 1</sub>)
If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with [[Trapezoidal rule|trapezoids]] and:
:<math>G_1 = 1 - \sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})</math>

is the resulting approximation for G. More accurate results can be obtained using other methods to [[Numerical integration|approximate the area]] B, such as approximating the Lorenz curve with a [[Simpson's rule|quadratic function]] across pairs of intervals or building an appropriately smooth approximation to the underlying distribution function that matches the known data. If the population mean and boundary values for each interval are also known, these can also often be used to improve the accuracy of the approximation.

The Gini coefficient calculated from a sample is a statistic, and its standard error, or confidence intervals for the population Gini coefficient, should be reported. These can be calculated using [[Resampling (statistics)#Bootstrap|bootstrap]] techniques, mathematically complicated and computationally demanding even in an era of fast computers.<ref>{{Cite web |last=Abdon |first=Mitch |date=2011-05-23 |title=Bootstrapping Gini |url=https://www.statadaily.com/bootstrapping-gini/ |access-date=2022-11-12 |website=Statadaily: Unsolicited advice for the interested |language=en-US}}</ref> Economist [[Tomson Ogwang]] made the process more efficient by setting up a "trick regression model" in which respective income variables in the sample are ranked, with the lowest income being allocated rank 1. The model then expresses the rank (dependent variable) as the sum of a constant ''A'' and a [[normal distribution|normal]] error term whose variance is inversely proportional to ''y''<sub>''k''</sub>:

:<math>k = A + \ N(0, s^{2}/y_k) </math>

Thus, ''G'' can be expressed as a function of the weighted [[Least-squares estimation|least squares estimate]] of the constant ''A'' and that this can be used to speed up the calculation of the [[Resampling (statistics)#Jackknife|jackknife]] estimate for the standard error. Economist David Giles argued that the [[standard error]] of the estimate of ''A'' can be used to derive the estimate of ''G'' directly without using a jackknife. This method only requires using ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the [[Jackknife resampling|jackknife]] with agreement improving with increasing sample size.{{sfnp|Giles|2004}}

However, it has been argued that this depends on the model's assumptions about the error distributions and the independence of error terms. These assumptions are often not valid for real data sets. There is still ongoing debate surrounding this topic.

[[Guillermina Jasso]]<ref>{{cite journal|last=Jasso|first=Guillermina|year=1979|title=On Gini's Mean Difference and Gini's Index of Concentration|journal=American Sociological Review|volume=44|issue=5|pages=867–870|jstor=2094535|doi=10.2307/2094535}}</ref> and [[Angus Deaton]]{{sfnp|Deaton|1997|p=139}} independently proposed the following formula for the Gini coefficient:

:<math>G = \frac{N+1}{N-1}-\frac{2}{N(N-1)\mu}(\sum_{i=1}^n P_iX_i)</math>

where <math>\mu</math> is mean income of the population, P<sub>i</sub> is the income rank P of person i, with income X, such that the richest person receives a rank of 1 and the poorest a rank of ''N''. This effectively gives higher weight to poorer people in the income distribution, which allows the Gini to meet the [[Income inequality metrics#Transfer principle|Transfer Principle]]. Note that the Jasso-Deaton formula rescales the coefficient so that its value is one if all the <math>X_i</math> are zero except one. Note however Allison's reply on the need to divide by N² instead.<ref>{{cite journal|title=Reply to Jasso|first=Paul D.|last=Allison|journal=American Sociological Review|volume=44|issue=5|year=1979|pages=870–872|jstor=2094536|doi=10.2307/2094536}}<!--|access-date=2 February 2015--></ref>

[[FAO]] explains another version of the formula.<ref name="fao gini">{{cite web|title=Inequality Analysis – The Gini Index|publisher=Food and Agriculture Organization, United Nations|first1=Lorenzo Giovanni|last1=Bellù|first2=Paolo|last2=Liberati|year=2006|url=http://www.fao.org/docs/up/easypol/329/gini_index_040EN.pdf|access-date=31 July 2012|archive-date=13 July 2017|archive-url=https://web.archive.org/web/20170713164057/http://www.fao.org/docs/up/easypol/329/gini_index_040en.pdf|url-status=dead}}</ref>

== Generalized inequality indices ==
{{See also|Generalized entropy index}}

The Gini coefficient and other standard inequality indices reduce to a common form. Perfect equality—the absence of inequality—exists when and only when the inequality ratio, <math>r_j = x_j / \overline{x}</math>, equals 1 for all j units in some population (for example, there is perfect income equality when everyone's income <math>x_j</math> equals the mean income <math>\overline{x}</math>, so that <math>r_j=1</math> for everyone). Measures of inequality, then, are measures of the average deviations of the <math>r_j=1</math> from 1; the greater the average deviation, the greater the inequality. Based on these observations the inequality indices have this common form:<ref>{{Cite journal |last=Firebaugh |first=Glenn |year=1999 |title=Empirics of World Income Inequality |journal=American Journal of Sociology |volume=104 |issue=6 |pages=1597–1630 |doi=10.1086/210218 |s2cid=154973184 }}. See also {{Cite book |chapter=Inequality: What it is and how it is measured |first=Glenn |last=Firebaugh |author-mask=3 |title=The New Geography of Global Income Inequality |location=Cambridge, MA |publisher=Harvard University Press |year=2003 |isbn=978-0-674-01067-3}}</ref>

:<math>\text{Inequality} = \sum_j p_j \, f(r_j), </math>

where ''p''<sub>''j''</sub> weights the units by their population share, and ''f''(''r''<sub>''j''</sub>) is a function of the deviation of each unit's ''r''<sub>''j''</sub> from 1, the point of equality. The insight of this generalized inequality index is that inequality indices differ because they employ different functions of the distance of the inequality ratios (the ''r''<sub>''j''</sub>) from 1.

== Of income distributions ==
{{See also|List of countries by income equality}}
{{Lorenz curve global income 2011.svg}}

Gini coefficients of income are calculated on a market income and a disposable income basis. The Gini coefficient on market income—sometimes referred to as a pre-tax Gini coefficient—is calculated on income before taxes and transfers. It measures inequality in income without considering the effect of taxes and social spending already in place in a country. The Gini coefficient on disposable income—sometimes referred to as the after-tax Gini coefficient—is calculated on income after taxes and transfers. It measures inequality in income after considering the effect of taxes and social spending already in place in a country.<ref name=OECD1 /><ref>{{cite journal|title=Applications of Lorenz Curves in Economic Analysis|first=N. C.|last=Kakwani|journal=Econometrica|volume=45|issue=3|date=April 1977|pages= 719–728|jstor=1911684|doi=10.2307/1911684}}</ref><ref name=imf2000>{{cite web|title=Income Distribution and Tax and Government Social Spending Policies in Developing Countries|publisher=International Monetary Fund|date=March 2000|last1=Chu|first1=Ke-young|last2=Davoodi|first2=Hamid|last3=Gupta|first3=Sanjeev|url=http://www.imf.org/external/pubs/ft/wp/2000/wp0062.pdf |archive-url=https://web.archive.org/web/20000830022345/http://www.imf.org/external/pubs/ft/wp/2000/wp0062.pdf |archive-date=2000-08-30 |url-status=live}}</ref>

For [[OECD]] countries over the 2008–2009 period, the Gini coefficient (pre-taxes and transfers) for a total population ranged between 0.34 and 0.53, with South Korea the lowest and Italy the highest. The Gini coefficient (after-taxes and transfers) for a total population ranged between 0.25 and 0.48, with Denmark the lowest and Mexico the highest. For the United States, the country with the largest population among OECD countries, the pre-tax Gini index was 0.49, and the after-tax Gini index was 0.38 in 2008–2009. The OECD average for total populations in OECD countries was 0.46 for the pre-tax income Gini index and 0.31 for the after-tax income Gini index.<ref name=OECD1 /><ref>{{cite web |url=http://www.eurofound.europa.eu/areas/qualityoflife/eurlife/index.php?template=3&radioindic=158&idDomain=3 |title=Monitoring quality of life in Europe – Gini index |work=Eurofound |date=26 August 2009 |url-status=dead |archive-url=https://web.archive.org/web/20081201193249/http://www.eurofound.europa.eu/areas/qualityoflife/eurlife/index.php?template=3&radioindic=158&idDomain=3 |archive-date=1 December 2008 |df=dmy-all }}</ref> Taxes and social spending that were in place in 2008–2009 period in OECD countries significantly lowered effective income inequality, and in general, "European countries—especially Nordic and Continental [[welfare states]]—achieve lower levels of income inequality than other countries."<ref>{{cite journal|title=The redistributive effect of social transfer programmes and taxes: A decomposition across countries|first1=Chen|last1=Wang|first2=Koen|last2=Caminada|first3=Kees|last3=Goudswaard |journal=International Social Security Review|volume=65|issue=3|pages=27–48|year=2012|doi=10.1111/j.1468-246X.2012.01435.x|s2cid=154029963|hdl=1887/3207160|hdl-access=free}}</ref>

Using the Gini can help quantify differences in [[Welfare spending|welfare]] and [[living wage|compensation]] policies and philosophies. However, it should be borne in mind that the Gini coefficient can be misleading when used to make political comparisons between large and small countries or those with different immigration policies (see [[#Limitations|limitations]] section).

The Gini coefficient for the entire world has been estimated by various parties to be between 0.61 and 0.68.<ref name=fao2009 /><ref name=undp10>{{cite book|title=The Real Wealth of Nations: Pathways to Human Development, 2010|publisher=United Nations Development Program|year=2011|pages=72–74|isbn= 978-0-230-28445-6|url=http://hdr.undp.org/en/media/HDR_2010_EN_Complete_reprint.pdf|archive-url=https://web.archive.org/web/20110429050250/http://hdr.undp.org/en/media/HDR_2010_EN_Complete_reprint.pdf|archive-date=29 April 2011|last1=Nations |first1=United }}</ref><ref>{{cite web|url=http://siteresources.worldbank.org/INTDECINEQ/Resources/PSBSutcliffe.pdf |archive-url=https://web.archive.org/web/20070621041257/http://siteresources.worldbank.org/INTDECINEQ/Resources/PSBSutcliffe.pdf |archive-date=2007-06-21 |url-status=live|title=Postscript to the article 'World inequality and globalization' (Oxford Review of Economic Policy, Spring 2004)
|first=Bob|last=Sutcliffe|date=April 2007|access-date=13 December 2007}}</ref> The graph shows the values expressed as a percentage in their historical development for a number of countries.
[[File:Gini since WWII.svg|thumb|center|upright=3.25|alt=The change in Gini indices has differed across countries. Some countries have change little over time, such as Belgium, Canada, Germany, Japan, and Sweden. Brazil has oscillated around a steady value. France, Italy, Mexico, and Norway have shown marked declines. China and the US have increased steadily. Australia grew to moderate levels before dropping. India sank before rising again. The UK and Poland stayed at very low levels before rising. Bulgaria had an increase of fits-and-starts. .svg alt text]]

=== Regional income Gini indices ===
According to UNICEF, Latin America and the Caribbean region had the highest net income Gini index in the world at 48.3, on an unweighted average basis in 2008. The remaining regional averages were: sub-Saharan Africa (44.2), Asia (40.4), Middle East and North Africa (39.2), Eastern Europe and Central Asia (35.4), and High-income Countries (30.9). Using the same method, the United States is claimed to have a Gini index of 36, while South Africa had the highest income Gini index score of 67.8.<ref name=unicef2011>{{cite web|title=Global Inequality: Beyond the Bottom Billion|publisher=UNICEF|date=April 2011|first1=Isabel|last1=Ortiz|first2=Matthew|last2=Cummins|page=26|url=http://www.unicef.org/socialpolicy/files/Global_Inequality_REVISED_-_5_July.pdf|access-date=30 July 2012|archive-date=12 August 2012|archive-url=https://web.archive.org/web/20120812232455/http://www.unicef.org/socialpolicy/files/Global_Inequality_REVISED_-_5_July.pdf|url-status=dead}}</ref>

=== World income Gini index since 1800s ===
Taking income distribution of all human beings, worldwide income inequality has been constantly increasing since the early 19th century (and will keep on increasing over the years) . There was a steady increase in the global income inequality Gini score from 1820 to 2002, with a significant increase between 1980 and 2002. This trend appears to have peaked and begun a reversal with rapid economic growth in emerging economies, particularly in the large populations of [[BRIC]] countries.<ref>{{cite journal|title=More or Less|journal=Finance & Development|date= September 2011|volume= 48|issue= 3|first=Branko|last=Milanovic|url=http://www.imf.org/external/pubs/ft/fandd/2011/09/milanovic.htm}}</ref>

The table below presents the estimated world income Gini coefficients over the last 200 years, as calculated by Milanovic.<ref>{{Cite web |last= Milanovic |first= Branko |year= 2009 |title= Global Inequality and the Global Inequality Extraction Ratio |publisher=World Bank|url=http://www-wds.worldbank.org/servlet/WDSContentServer/WDSP/IB/2009/09/09/000158349_20090909092401/Rendered/PDF/WPS5044.pdf |archive-url=https://web.archive.org/web/20131111142720/http://www-wds.worldbank.org/servlet/WDSContentServer/WDSP/IB/2009/09/09/000158349_20090909092401/Rendered/PDF/WPS5044.pdf |archive-date=2013-11-11 |url-status=live}}</ref> 
{| class="wikitable" style="text-align: center;"
|+ Income Gini coefficient - World, 1820–2005
|-
! Year !! World Gini coefficients<ref name=fao2009 /><ref name=unicef2011 /><ref>{{cite web|title=Riding the Elephants: The Evolution of World Economic Growth and Income Distribution at the End of the Twentieth Century (1980–2000)|first1=Albert|last1=Berry |first2=John|last2=Serieux |publisher=United Nations (DESA Working Paper No. 27)|date=September 2006|url=https://www.un.org/esa/desa/papers/2006/wp27_2006.pdf |archive-url=https://web.archive.org/web/20090217064745/http://www.un.org/esa/desa/papers/2006/wp27_2006.pdf |archive-date=2009-02-17 |url-status=live}}</ref>
|-
| 1820 || 0.43
|-
| 1850 || 0.53
|-
| 1870 || 0.56
|-
| 1913 || 0.61
|-
| 1929 || 0.62
|-
| 1950 || 0.64
|-
| 1960 || 0.64
|-
| 1980 || 0.66
|-
| 2002 || 0.71
|-
| 2005 || 0.68
|}

More detailed data from similar sources plots a continuous decline since 1988.  This is attributed to [[globalization]] increasing incomes for billions of poor people, mostly in countries like China and India.  Developing countries like Brazil have also improved basic services like health care, education, and sanitation; others like Chile and Mexico have enacted more [[progressive tax]] policies.<ref>{{cite web|url=https://www.npr.org/sections/goatsandsoda/2017/01/25/511594991/what-the-stat-about-the-8-richest-men-doesnt-tell-us-about-inequality|title=What The Stat About The 8 Richest Men Doesn't Tell Us About Inequality| website=NPR | date=25 January 2017 | last1=Gharib | first1=Malaka }}</ref>

{| class="wikitable" style="text-align: center;"
|+ Income Gini coefficient - World, 1988–2013
|-
!Year
!World Gini coefficients<ref>{{cite web| title= Poverty and Prosperity 2016 / Taking on Inequality |url=https://openknowledge.worldbank.org/bitstream/handle/10986/25078/9781464809583.pdf |archive-url=https://web.archive.org/web/20161115132032/https://openknowledge.worldbank.org/bitstream/handle/10986/25078/9781464809583.pdf |archive-date=2016-11-15 |url-status=live |author=World Bank|author-link=World Bank }}. Figure O.10 
Global Inequality, 1988–2013</ref>
|-
|1988 || 0.80
|-
|1993 || 0.76
|-
|1998 || 0.74
|-
|2003 || 0.72
|-
|2008 || 0.70
|-
|2013 || 0.65
|}

== Of social development ==
The Gini coefficient is widely used in fields as diverse as sociology, economics, health science, ecology, engineering, and agriculture.<ref name="Sadras 2004 303–310">{{cite journal |last1=Sadras |first1=V. O. |last2=Bongiovanni |first2=R. |year=2004 |title=Use of Lorenz curves and Gini coefficients to assess yield inequality within paddocks |journal=Field Crops Research |volume=90 |issue=2–3 |pages=303–310 |doi=10.1016/j.fcr.2004.04.003 |bibcode=2004FCrRe..90..303S }}</ref> For example, in social sciences and economics, in addition to income Gini coefficients, scholars have published education Gini coefficients and opportunity Gini coefficients.

=== Education ===
Education Gini index estimates the inequality in education for a given population.<ref>{{cite book|first1=Vinod|last1=Thomas|first2=Yan|last2=Wang|first3=Xibo|last3=Fan|title=Measuring Education Inequality: Gini Coefficients of Education |date=January 2001|publisher=The World Bank|doi=10.1596/1813-9450-2525|url=http://www-wds.worldbank.org/servlet/WDSContentServer/WDSP/IB/2001/02/17/000094946_01020605310354/Rendered/PDF/multi_page.pdf|url-status=dead|archive-url=https://web.archive.org/web/20130605143955/http://www-wds.worldbank.org/servlet/WDSContentServer/WDSP/IB/2001/02/17/000094946_01020605310354/Rendered/PDF/multi_page.pdf|archive-date=5 June 2013|series=Policy Research Working Papers|hdl=10986/19738|citeseerx=10.1.1.608.6919|s2cid=6069811}}</ref> It is used to discern trends in social development through educational attainment over time. A study across 85 countries by three [[World Bank]] economists, Vinod Thomas, Yan Wang, and Xibo Fan, estimated Mali had the highest education Gini index of 0.92 in 1990 (implying very high inequality in educational attainment across the population), while the United States had the lowest education inequality Gini index of 0.14. Between 1960 and 1990, China, India and South Korea had the fastest drop in education inequality Gini Index. They also claim education Gini index for the United States slightly increased over the 1980–1990 period.

Though India's education Gini Index has been falling from 1960 through 1990, most of the population still has not received any education, while 10 percent of the population received more than 40% of the total educational hours in the nation. This means that a large portion of capable children in the country are not receiving the support necessary to allow them to become positive contributors to society. This will lead to a deadweight loss to the national society because there are many people who are underdeveloped and underutilized.<ref>{{Cite book |last1=Thomas |first1=Vinod |url=https://books.google.com/books?id=cVkVi5bzEQkC&dq=gini+coefficient&pg=PA3 |title=Measuring Education Inequality: Gini Coefficients of Education |last2=Wang |first2=Yan |last3=Fan |first3=Xibo |date=2001 |publisher=World Bank Publications |language=en}}</ref>

=== Opportunity ===
Similar in concept to the Gini income coefficient, the Gini opportunity coefficient measures inequality in opportunities.<ref name=roemer06>{{cite report|title=Economic development as opportunity equalization|first=John E.|last=Roemer|date=September 2006|publisher=Yale University|ssrn=931479|citeseerx=10.1.1.403.4725}}</ref><ref>{{cite journal|title=Generalized Gini Indices of Equality of Opportunity|last=Weymark|first=John|journal=Journal of Economic Inequality|volume=1|year=2003|pages=5–24|doi=10.1023/A:1023923807503|issue=1|s2cid=133596675}}</ref><ref>{{cite web|title=Measurement of Inequality in Human Development – A Review|last=Kovacevic|first=Milorad|publisher=United Nations Development Program|date=November 2010|url=http://hdr.undp.org/en/reports/global/hdr2010/papers/HDRP_2010_35.pdf|archive-url=https://web.archive.org/web/20110923141455/http://hdr.undp.org/en/reports/global/hdr2010/papers/HDRP_2010_35.pdf|archive-date=23 September 2011}}</ref> The concept builds on [[Amartya Sen]]'s suggestion<ref>{{cite journal|title=The contributions of Amartya Sen to Welfare Economics|first=Anthony B.|last=Atkinson|journal=The Scandinavian Journal of Economics|volume=101|issue=2|pages=173–190|year=1999|url=http://ias7.berkeley.edu:80/academics/courses/center/fall2007/sehnbruch/atkinson%201998%20contributions%20of%20sen%20to%20welf%20economics.pdf|archiveurl=https://web.archive.org/web/20140513233332/http://ias7.berkeley.edu:80/academics/courses/center/fall2007/sehnbruch/atkinson%201998%20contributions%20of%20sen%20to%20welf%20economics.pdf |archivedate=2014-05-13|doi=10.1111/1467-9442.00151|jstor=3440691}}{{cbignore|bot=medic}}</ref> that inequality coefficients of social development should be premised on the process of enlarging people's choices and enhancing their capabilities, rather than on the process of reducing income inequality. Kovacevic, in a review of the Gini opportunity coefficient, explained that the coefficient estimates how well a society enables its citizens to achieve success in life where the success is based on a person's choices, efforts and talents, not their background defined by a set of predetermined circumstances at birth, such as gender, race, place of birth, parent's income and circumstances beyond the control of that individual.

In 2003, Roemer<ref name=roemer06 /><ref>{{cite journal|last1=Roemer|first1=John E.|title=To what extent do fiscal regimes equalize opportunities for income acquisition among citizens?|journal=Journal of Public Economics|volume=87|issue=3–4|date=March 2003|pages=539–565|doi=10.1016/S0047-2727(01)00145-1|last2=Aaberge|first2=Rolf|last3=Colombino|first3=Ugo|last4=Fritzell|first4=Johan|last5=Jenkins|first5=Stephen P|last6=Lefranc|first6=Arnaud|last7=Marx|first7=Ive|last8=Page|first8=Marianne|last9=Pommer|first9=Evert|display-authors=1|citeseerx=10.1.1.414.6220}}</ref> reported Italy and Spain exhibited the largest opportunity inequality Gini index amongst advanced economies.

=== Income mobility ===
In 1978, [[Anthony Shorrocks]] introduced a measure based on income Gini coefficients to estimate income mobility.<ref>{{Cite journal | last = Shorrocks | first = Anthony | title = Income inequality and income mobility | journal = Journal of Economic Theory | volume = 19 | issue = 2 | pages = 376–393 | doi = 10.1016/0022-0531(78)90101-1 | date = December 1978 }}</ref> This measure, generalized by Maasoumi and Zandvakili,<ref>{{cite journal|first1=Esfandiar|last1=Maasoumi|first2=Sourushe|last2=Zandvakili| year=1986|title=A class of generalized measures of mobility with applications|journal=Economics Letters|volume= 22|issue=1|pages= 97–102|doi=10.1016/0165-1765(86)90150-3}}</ref> is now generally referred to as [[Shorrocks index]], sometimes as Shorrocks mobility index or Shorrocks rigidity index. It attempts to estimate whether the income inequality Gini coefficient is permanent or temporary and to what extent a country or region enables economic mobility to its people so that they can move from one (e.g., bottom 20%) income quantile to another (e.g., middle 20%) over time. In other words, the Shorrocks index compares inequality of short-term earnings, such as the annual income of households, to inequality of long-term earnings, such as 5-year or 10-year total income for the same households.

Shorrocks index is calculated in several different ways, a common approach being from the ratio of income Gini coefficients between short-term and long-term for the same region or country.<ref name=kss2010>{{cite journal|title=Earnings Inequality and Mobility in the United States: Evidence from Social Security Data Since 1937|first1=Wojciech|last1=Kopczuk |first2=Emmanuel|last2=Saez |first3=Jae|last3=Song |journal=The Quarterly Journal of Economics|year=2010|volume= 125|issue=1|pages= 91–128|doi=10.1162/qjec.2010.125.1.91|jstor=40506278|url=http://emlab.berkeley.edu/~saez/kopczuk-saez-songQJE09SSA.pdf |archive-url=https://web.archive.org/web/20130513122750/http://emlab.berkeley.edu/~saez/kopczuk-saez-songQJE09SSA.pdf |archive-date=2013-05-13 |url-status=live}}</ref>

A 2010 study using social security income data for the United States since 1937 and Gini-based Shorrock's indices concludes that income mobility in the United States has had a complicated history, primarily due to the mass influx of women into the American labor force after World War II. Income inequality and income mobility trends have been different for men and women workers between 1937 and the 2000s. When men and women are considered together, the Gini coefficient-based Shorrocks index trends imply long-term income inequality has been substantially reduced among all workers, in recent decades for the United States.<ref name=kss2010 /> Other scholars, using just 1990s data or other short periods have come to different conclusions.<ref>{{cite journal|title=Cross-national Differences in Income Mobility: Evidence from Canada, the United States, Great Britain and Germany |first=Wen-Hao|last=Chen|s2cid=62886186|journal=Review of Income and Wealth|volume=55|issue=1|pages=75–100|date= March 2009|doi=10.1111/j.1475-4991.2008.00307.x}}</ref> For example, Sastre and Ayala conclude from their study of income Gini coefficient data between 1993 and 1998 for six developed economies that France had the least income mobility, Italy the highest, and the United States and Germany intermediate levels of income mobility over those five years.<ref>{{cite web|title=Europe vs. The United States: Is There a Trade-Off Between Mobility and Inequality?|first1=Mercedes|last1=Sastre|first2=Luis|last2=Ayala |publisher=Institute for Social and Economic Research, University of Essex|url=http://www.ucm.es/info/econeuro/documentos/documentos/dt192002.pdf |archive-url=https://web.archive.org/web/20060612200520/http://www.ucm.es/info/econeuro/documentos/documentos/dt192002.pdf |archive-date=2006-06-12 |url-status=live|year=2002}}</ref>

== Features ==
{{Expand section|date=March 2023}}
The Gini coefficient has features that make it useful as a measure of dispersion in a population, and inequalities in particular.<ref name="fao gini" /> The coefficient ranges from 0, for perfect equality, to 1, indicating perfect inequality. The Gini is based on the comparison of cumulative proportions of the population against cumulative proportions of income they receive.<ref>{{Cite web |title=Inequality - Income inequality - OECD Data |url=http://data.oecd.org/inequality/income-inequality.htm |access-date=2023-06-02 |website=theOECD |language=en}}</ref>

== Limitations ==

===Relative, not absolute===
The Gini coefficient is a relative measure. The Gini coefficient of a developing country can rise (due to increasing inequality of income) even when the number of people in absolute poverty decreases.<ref>{{cite web|title=Dramatic Poverty Reduction in the Third World: Prospects and Needed Action|first=John W.|last=Mellor|publisher=International Food Policy Research Institute|date=2 June 1989|pages=18–20|url=http://pdf.usaid.gov/pdf_docs/PNABK503.pdf |archive-url=https://web.archive.org/web/20120803160551/http://pdf.usaid.gov/pdf_docs/PNABK503.pdf |archive-date=2012-08-03 |url-status=dead}}</ref> This is because the Gini coefficient measures relative, not absolute, wealth.

Gini coefficients are simple, and this simplicity can lead to oversights and can confuse the comparison of different populations; for example, while both Bangladesh (per capita income of $1,693) and the Netherlands (per capita income of $42,183) had an income Gini coefficient of 0.31 in 2010,<ref name=undp2010a>{{cite web|title=The Real Wealth of Nations: Pathways to Human Development (2010 Human Development Report – see Stat Tables)|pages=152–156|publisher=United Nations Development Program|year=2011|url=http://hdr.undp.org/en/reports/global/hdr2010/chapters/}}</ref> the quality of life, economic opportunity and absolute income in these countries are very different, i.e. countries may have identical Gini coefficients, but differ greatly in wealth. Basic necessities may be available to all in a developed economy, while in an undeveloped economy with the same Gini coefficient, basic necessities may be unavailable to most or unequally available due to lower absolute wealth.

===Mathematical limitations===
Gini has some mathematical limitations as well. It is not additive and different sets of people cannot be averaged to obtain the Gini coefficient of all the people in the sets.

{| class="wikitable" style="float: right; margin-left:1em;"
|+ Table A. Different income distributions with the same Gini index<ref name="fao gini"/>
|-
! style=max-width:4em | Household group !! style=max-width:5em | Country A annual income ($) !! style=max-width:5em | Country B annual income ($)
|-
| 1 || 20,000 || 9,000
|-
| 2 || 30,000 || 40,000
|-
| 3 || 40,000 || 48,000
|-
| 4 || 50,000 || 48,000
|-
| 5 || 60,000 || 55,000
|-
| Total income || $200,000 || $200,000
|-
| Country's Gini || '''0.2''' || '''0.2'''
|}

Even when the total income of a population is the same, in certain situations two countries with different income distributions can have the same Gini index (e.g. cases when income Lorenz Curves cross).<ref name="fao gini"/> Table A illustrates one such situation. Both countries have a Gini coefficient of 0.2, but the average income distributions for household groups are different. As another example, in a population where the lowest 50% of individuals have no income, and the other 50% have equal income, the Gini coefficient is 0.5; whereas for another population where the lowest 75% of people have 25% of income and the top 25% have 75% of the income, the Gini index is also 0.5. Economies with similar incomes and Gini coefficients can have very different income distributions. Bellù and Liberati claim that ranking income inequality between two populations is not always possible based on their Gini indices.<ref name=Maio2007>{{cite journal|title=Income inequality measures|first=Fernando G.|last=De Maio|journal=Journal of Epidemiology and Community Health|year=2007|volume=61|issue=10|pages=849–852|doi=10.1136/jech.2006.052969|pmid=17873219|pmc=2652960}}</ref> Similarly, computational social scientist Fabian Stephany illustrates that income inequality within the population, e.g., in specific socioeconomic groups of same age and education, also remains undetected by conventional Gini indices.<ref>{{Cite journal |last=Stephany |first=Fabian |date=2017-12-01 |title=Who are Your Joneses? Socio-Specific Income Inequality and Trust |url=https://doi.org/10.1007/s11205-016-1460-9 |journal=Social Indicators Research |language=en |volume=134 |issue=3 |pages=877–898 |doi=10.1007/s11205-016-1460-9 |issn=1573-0921 |pmc=5684274 |pmid=29187771}}</ref>

===Income Gini can conceal wealth inequality===
A Gini index does not contain information about absolute national or personal incomes. Populations can simultaneously have very low income Gini indices and very high wealth Gini indexes. By measuring inequality in income, the Gini ignores the differential efficiency of the use of household income. By ignoring wealth (except as it contributes to income), the Gini can create the appearance of inequality when the people compared are at different stages in their life. Wealthy countries such as Sweden can show a low Gini coefficient for the disposable income of 0.31, thereby appearing equal, yet have a very high Gini coefficient for wealth of 0.79 to 0.86, suggesting an extremely unequal wealth distribution in its society.<ref>{{cite journal|title=Inequality Trends in Sweden 1978–2004|first1=David|last1=Domeij|first2=Martin|last2=Flodén|journal= Review of Economic Dynamics|volume=13|issue=1|year=2010|pages=179–208|doi=10.1016/j.red.2009.10.005|citeseerx=10.1.1.629.9417}}</ref><ref>{{cite web|title=Accounting for Swedish wealth inequality|first1=David|last1=Domeij|first2=Paul|last2=Klein|date=January 2000|url=http://fmwww.bc.edu/repec/es2000/0883.pdf|archive-url=https://web.archive.org/web/20030519155531/https://www.econometricsociety.org/meetings/wc00/pdf/0883.pdf|archive-date=May 19, 2003}}</ref> These factors are not assessed in income-based Gini.

===Country size and granularity bias===
Gini index has a downward-bias for small populations.<ref>{{cite journal|title=The Small-Sample Bias of the Gini Coefficient: Results and Implications for Empirical Research|journal=The Review of Economics and Statistics|first=George|last=Deltas|date=February 2003|volume= 85|issue=1|pages=226–234|doi=10.1162/rest.2003.85.1.226|jstor=3211637|s2cid=57572560}}</ref> Counties or states or countries with small populations and less diverse economies will tend to report small Gini coefficients. For economically diverse large population groups, a much higher coefficient is expected than for each of its regions. For example, taking the world economy as a whole and income distribution for all human beings, different scholars estimate the global Gini index to range between 0.61 and 0.68.<ref name=fao2009 /><ref name=undp10 />
As with other inequality coefficients, the Gini coefficient is influenced by the [[granularity]] of the measurements. For example, five 20% quantiles (low granularity) will usually yield a lower Gini coefficient than twenty 5% quantiles (high granularity) for the same distribution. Philippe Monfort has shown that using inconsistent or unspecified granularity limits the usefulness of Gini coefficient measurements.<ref>{{cite web|title=Convergence of EU regions: Measures and evolution|first=Philippe|last=Monfort|publisher=European Union – Europa|page=6|year=2008|url=http://ec.europa.eu/regional_policy/sources/docgener/work/200801_convergence.pdf |archive-url=https://web.archive.org/web/20120803160544/http://ec.europa.eu/regional_policy/sources/docgener/work/200801_convergence.pdf |archive-date=2012-08-03 |url-status=live}}</ref>

===Changes in population===
Changing income inequality, measured by Gini coefficients, can be due to structural changes in a society such as growing population (increased birth rates, aging populations, emigration, immigration) and income mobility.<ref name=Kwok10>{{cite web|title=Income Distribution of Hong Kong and the Gini Coefficient|author=KWOK Kwok Chuen|year=2010|publisher=The Government of Hong Kong, China|url=http://www.eabfu.gov.hk/en/pdf/income.pdf|archive-url=https://web.archive.org/web/20101227043822/http://www.eabfu.gov.hk/en/pdf/income.pdf|archive-date=27 December 2010}}</ref>

Another limitation of the Gini coefficient is that it is not a proper measure of [[egalitarianism]], as it only measures income dispersion. For example, suppose two equally egalitarian countries pursue different [[immigration law|immigration policies]]. In that case, the country accepting a higher proportion of low-income or impoverished migrants will report a higher Gini coefficient and, therefore, may exhibit more income inequality.

===Household vs individual===
{| class="wikitable" style="text-align:center; float: right; margin-left:1em;"
|+ Table B. Same income distributions, but different Gini Index
|-
! style=max-width:4em | Household number !! style=max-width:5em | Country  Annual Income ($) !! style=max-width:5em |Household combined number !! style=max-width:5em |Country A combined Annual Income ($)
|-
| 1 || 20,000 || rowspan="2"  style="text-align: center;" | 1 & 2 || rowspan="2" style="text-align: center;" | 50,000
|-
| 2 || 30,000
|-
| 3 || 40,000 || rowspan="2"  style="text-align: center;" | 3 & 4 || rowspan="2" style="text-align: center;" | 90,000
|-
| 4 || 50,000
|-
| 5 || 60,000 || rowspan="2"  style="text-align: center;" | 5 & 6 || rowspan="2" style="text-align: center;" | 130,000
|-
| 6 || 70,000
|-
| 7 || 80,000 || rowspan="2" style="text-align: center;" | 7 & 8 || rowspan="2" style="text-align: center;" | 170,000
|-
| 8 || 90,000
|-
| 9 || 120,000 || rowspan="2"  style="text-align: center;" | 9 & 10 || rowspan="2"  style="text-align: center;" | 270,000
|-
| 10 || 150,000
|-
| Total Income || $710,000 ||  || $710,000
|-
| Country's Gini || '''0.303''' || ||'''0.293'''
|}
The Gini coefficient measure gives different results when applied to individuals instead of households, for the same economy and same income distributions. If household data is used, the measured value of income Gini depends on how the household is defined. The comparison is not meaningful when different populations are not measured with consistent definitions. Furthermore, changes to the household income Gini can be driven by changes in household formation, such as increased divorce rates or [[extended family]] households splitting into [[Nuclear family|nuclear families]].

Deininger and [[Lyn Squire|Squire]] (1996) show that the income Gini coefficient based on individual income rather than household income is different. For example, for the United States, they found that the individual income-based Gini index was 0.35, while for France, 0.43. According to their individual-focused method, in the 108 countries they studied, South Africa had the world's highest Gini coefficient at 0.62, Malaysia had Asia's highest Gini coefficient at 0.5, Brazil the highest at 0.57 in Latin America and the Caribbean region, and Turkey the highest at 0.5 in OECD countries.<ref>{{cite journal |last1=Deininger |first1=K. |last2=Squire |first2=L. |title=A New Data Set Measuring Income Inequality |journal=The World Bank Economic Review |date=September 1996 |volume=10 |issue=3 |pages=565–591 |doi=10.1093/wber/10.3.565 }}</ref>

Billionaire [[Thomas Kwok]] claimed the income Gini coefficient for Hong Kong has been high (0.434 in 2010<ref name=undp2010a />), in part because of structural changes in its population. Over recent decades, Hong Kong has witnessed increasing numbers of small households, elderly households, and elderly living alone. The combined income is now split into more households. Many older people live separately from their children in Hong Kong. These social changes have caused substantial changes in household income distribution. The income Gini coefficient, claims Kwok, does not discern these structural changes in its society.<ref name=Kwok10 /> Household money income distribution for the United States, summarized in Table C of this section, confirms that this issue is not limited to just Hong Kong. According to the US Census Bureau, between 1979 and 2010, the population of the United States experienced structural changes in overall households; the income for all income brackets increased in inflation-adjusted terms, household income distributions shifted into higher income brackets over time, while the income Gini coefficient increased.<ref name=uscb2011 /><ref name=cbo.p.10>[http://www.cbo.gov/doc.cfm?index=12485 Congressional Budget Office: Trends in the Distribution of Household Income Between 1979 and 2007]. October 2011. see pp. i–x, with definitions on ii–iii</ref>

{| class="wikitable" style="text-align:center; float: left; margin-right:1em;"
|+ Table C. Household money income distributions and Gini Index, US<ref name=uscb2011>{{cite web|title=Income, Poverty, and Health Insurance Coverage in the United States: 2010 (see Table A-2)|date=September 2011|publisher=Census Bureau, Dept of Commerce, United States|url=https://www.census.gov/prod/2011pubs/p60-239.pdf |archive-url=https://web.archive.org/web/20110923022827/http://www.census.gov/prod/2011pubs/p60-239.pdf |archive-date=2011-09-23 |url-status=live}}</ref>
|-
! style=max-width:6em | Income bracket (in 2010 adjusted dollars)!! style=max-width:4em | % of Population 1979 !! style=max-width:4em | % of Population 2010
|-
| Under $15,000 || 14.6% || 13.7%
|-
| $15,000 – $24,999 || 11.9% || 12.0%
|-
| $25,000 – $34,999 || 12.1% || 10.9%
|-
| $35,000 – $49,999 || 15.4% || 13.9%
|-
| $50,000 – $74,999 || 22.1% || 17.7%
|-
| $75,000 – $99,999 || 12.4% || 11.4%
|-
| $100,000 – $149,999 || 8.3% || 12.1%
|-
| $150,000 – $199,999 || 2.0% || 4.5%
|-
| $200,000 and over || 1.2% || 3.9%
|-
| Total Households || 80,776,000 || 118,682,000
|-
| United States' Gini on pre-tax basis || '''0.404''' || '''0.469'''
|}

===Instantaneous inequality vs lifetime inequality===
The Gini coefficient is unable to discern the effects of structural changes in populations.<ref name=Kwok10 /> Expanding on the importance of life-span measures, the Gini coefficient as a point-estimate of equality at a certain time ignores life-span changes in income. Typically, increases in the proportion of young or old members of a society will drive apparent changes in equality simply because people generally have lower incomes and wealth when they are young than when they are old. Because of this, factors such as age distribution within a population and mobility within income classes can create the appearance of inequality when none exist, taking into account demographic effects. Thus a given economy may have a higher Gini coefficient at any timepoint compared to another, while the Gini coefficient calculated over individuals' lifetime income is lower than the apparently more equal (at a given point in time) economy's.{{Clarify|date=September 2022}}<ref name=blomq81>{{Cite journal |first=N. |last=Blomquist |s2cid=154519005 |year=1981 |title=A comparison of distributions of annual and lifetime income: Sweden around 1970 |journal=Review of Income and Wealth |volume=27 |issue=3 |pages=243–264 |doi=10.1111/j.1475-4991.1981.tb00227.x}}</ref> Essentially, what matters is not just inequality in any particular year but the distribution composition over time.

===Benefits and income in kind===
Inaccuracies in assign monetary value to [[income in kind]] reduce the accuracy of Gini as a measurement of true inequality.

While taxes and cash transfers are relatively straightforward to account for, other government benefits can be difficult to value. Benefits such as subsidized housing, medical care, and education are difficult to value objectively, as it depends on the quality and extent of the benefit. In absence of a free market, valuing these income transfers as household income is subjective. The theoretical model of the Gini coefficient is limited to accepting correct or incorrect subjective assumptions.

In subsistence-driven and [[informal economies]], people may have significant income in other forms than money, for example, through [[subsistence farming]] or [[barter]]ing. These forms of income tend to accrue to poor segments of populations in emerging and transitional economy countries such as those in sub-Saharan Africa, Latin America, Asia, and Eastern Europe. Informal economy accounts for over half of global employment and as much as 90 percent of employment in some of the poorer sub-Saharan countries with high official Gini inequality coefficients. Schneider et al., in their 2010 study of 162 countries,<ref>{{cite journal|first1=Friedrich|last1=Schneider|first2=Andreas|last2=Buehn|first3=Claudio E.|last3=Montenegro|s2cid=56060172|year=2010|title= New Estimates for the Shadow Economies all over the World|journal=International Economic Journal|volume=24|issue=4|pages=443–461|doi=10.1080/10168737.2010.525974|hdl=10986/4929}}</ref> report about 31.2%, or about $20 trillion, of world's [[Gross domestic product|GDP]] is informal. In developing countries, the informal economy predominates for all income brackets except the richer, urban upper-income bracket populations. Even in developed economies, 8% (United States) to 27% (Italy) of each nation's GDP is informal. The resulting informal income predominates as a livelihood activity for those in the lowest income brackets.<ref>{{cite book|title=The Informal Economy|publisher=International Institute for Environment and Development, United Kingdom|year=2011|url=http://pubs.iied.org/pdfs/15515IIED.pdf |archive-url=https://web.archive.org/web/20120803160544/http://pubs.iied.org/pdfs/15515IIED.pdf |archive-date=2012-08-03 |url-status=live|isbn=978-1-84369-822-7}}</ref> The value and distribution of the incomes from informal or underground economy is difficult to quantify, making true income Gini coefficients estimates difficult.<ref name=mfeld>{{cite web|title=Is income inequality really the problem? (Overview)|first=Martin|last=Feldstein|date=August 1998|publisher=US Federal Reserve|url=http://www.kc.frb.org/publicat/sympos/1998/S98feldstein.pdf|access-date=2 August 2012|archive-date=3 August 2012|archive-url=https://web.archive.org/web/20120803160558/http://www.kc.frb.org/publicat/sympos/1998/S98feldstein.pdf|url-status=dead}}</ref><ref name=tandw>{{cite book|title=Principles of Microeconomics: Global Financial Crisis Edition|first1=John|last1=Taylor|first2=Akila|last2=Weerapana|year=2009|isbn=978-1-4390-7821-1|pages=416–418|publisher=Cengage Learning }}</ref> Different assumptions and quantifications of these incomes will yield different Gini coefficients.<ref>{{cite journal|title=Income Inequality and the Informal Economy in Transition Economies|first1=J. Barkley Jr. |last1=Rosser|first2=Marina V.|last2=Rosser |first3=Ehsan|last3=Ahmed |s2cid=49552052 |journal=Journal of Comparative Economics|date=March 2000|volume= 28|issue=1|pages=156–171|doi=10.1006/jcec.2000.1645}}</ref><ref>{{cite web|title=Earnings inequality and the informal economy: evidence from Serbia|first1=Gorana|last1=Krstić |first2=Peter|last2=Sanfey |publisher=European Bank for Reconstruction and Development|date=February 2010|url=http://www.ebrd.com/downloads/research/economics/workingpapers/wp0114.pdf |archive-url=https://web.archive.org/web/20120803160550/http://www.ebrd.com/downloads/research/economics/workingpapers/wp0114.pdf |archive-date=2012-08-03 |url-status=live}}</ref><ref>{{cite report|title=The Size of the Shadow Economies of 145 Countries all over the World: First Results over the Period 1999 to 2003|first=Friedrich|last=Schneider|date=December 2004|ssrn=636661|hdl=10419/20729}}</ref>

== Alternatives ==
Given the limitations of the Gini coefficient, other statistical methods are used in combination or as an alternative measure of population dispersity. For example, ''entropy measures'' are frequently used (e.g. the [[Atkinson index]] or the [[Theil Index]] and [[Mean log deviation]] as special cases of the [[generalized entropy index]]). These measures attempt to compare the distribution of resources by intelligent agents in the market with a maximum [[information entropy|entropy]] [[random distribution]], which would occur if these agents acted like non-interacting particles in a closed system following the laws of statistical physics.

The Ortego two-parameter model<ref name="ortega,1991">{{cite journal | last=Ortega | first=P. | last2=Martín | first2=G. | last3=Fernández | first3=A. | last4=Ladoux | first4=M. | last5=García | first5=A. | title=A NEW FUNCTIONAL FORM FOR ESTIMATING LORENZ CURVES | journal=Review of Income and Wealth | volume=37 | issue=4 | date=1991 | issn=0034-6586 | doi=10.1111/j.1475-4991.1991.tb00383.x | pages=447–452}}</ref> may be superior to the GINI index.<ref name="Blesch,2022">{{cite journal | last=Blesch | first=Kristin | last2=Hauser | first2=Oliver P. | last3=Jachimowicz | first3=Jon M. | title=Measuring inequality beyond the Gini coefficient may clarify conflicting findings | journal=Nature Human Behaviour | volume=6 | issue=11 | date=2022 | issn=2397-3374 | pmid=36038775 | pmc=7614289 | doi=10.1038/s41562-022-01430-7 | pages=1525–1536}}</ref>

== Relation to other statistical measures ==
There is a summary measure of the diagnostic ability of a binary classifier system that is also called the ''Gini coefficient'', which is defined as twice the area between the [[receiver operating characteristic]] (ROC) curve and its diagonal.  It is related to the [[Receiver operating characteristic#Area under the curve|AUC]] ([[Integral|Area Under]] the ROC Curve) measure of performance given by <math>AUC = (G+1)/2</math><ref name=hand>{{cite journal |last1=Hand |first1=David J. |last2=Till |first2=Robert J. |title=A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems |journal=Machine Learning |date=2001 |volume=45 |issue=2 |pages=171–186 |doi=10.1023/A:1010920819831 |doi-access=free }}</ref>   and to [[Mann–Whitney U]].  Although both Gini coefficients are defined as areas between certain curves and share certain properties, there is no simple direct relationship between the Gini coefficient of statistical dispersion and the Gini coefficient of a classifier.
 
The Gini index is also related to the Pietra index — both of which measure statistical heterogeneity and are derived from the Lorenz curve and the diagonal line.<ref>{{cite journal|first1=Iddo I.|last1=Eliazar |first2=Igor M.|last2=Sokolov |year=2010 |title=Measuring statistical heterogeneity: The Pietra index|journal=Physica A: Statistical Mechanics and Its Applications|volume=389|issue= 1|pages= 117–125|doi= 10.1016/j.physa.2009.08.006|bibcode=2010PhyA..389..117E}}</ref><ref>{{cite journal |last1=Lee |first1=Wen-Chung |title=Probabilistic analysis of global performances of diagnostic tests: interpreting the Lorenz curve-based summary measures |journal=Statistics in Medicine |date=28 February 1999 |volume=18 |issue=4 |pages=455–471 |doi=10.1002/(sici)1097-0258(19990228)18:4<455::aid-sim44>3.0.co;2-a |pmid=10070686 }}</ref><ref name="McDonald1974"/>

In certain fields such as ecology, inverse Simpson's index <math>1/\lambda</math> is used to quantify diversity, and this should not be confused with the [[Diversity index#Simpson index|Simpson index]] <math>\lambda</math>. These indicators are related to Gini. The inverse Simpson index increases with diversity, unlike the Simpson index and Gini coefficient, which decrease with diversity. The Simpson index is in the range [0, 1], where 0 means maximum and 1 means minimum diversity (or heterogeneity). Since diversity indices typically increase with increasing heterogeneity, the Simpson index is often transformed into inverse Simpson, or using the complement <math>1 - \lambda</math>, known as the Gini-Simpson Index.<ref>{{cite journal|title=The Measurement of Species Diversity|first=Robert K.|last=Peet|s2cid=83517584|journal=Annual Review of Ecology and Systematics|volume=5|year=1974 |issue=1 |pages= 285–307|jstor=2096890|doi=10.1146/annurev.es.05.110174.001441|bibcode=1974AnRES...5..285P }}</ref>

The [[Lorenz curve]] is another method of graphical representation of wealth distribution. It was developed 9 years before the Gini coefficient, which quantifies the extent to which the Lorenz curve deviates from the perfect equality line (with [[slope]] of 1). The [[Hoover index]] (also known as Robin Hood index) presents the percentage of total population's income that would have to be redistributed to make the Gini coefficient equal to 0 (perfect equality).<ref>{{Cite web |title=Hoover Index |url=https://corporatefinanceinstitute.com/resources/economics/hoover-index/ |access-date=2024-04-28 |website=Corporate Finance Institute |language=en-US}}</ref>

== Gini coefficients for pre-modern societies ==
In recent decades, researchers have attempted to estimate Gini coefficients for pre-20th century societies. In the absence of household income surveys and income taxes, scholars have relied on proxy variables. These include wealth taxes in medieval European city states, patterns of landownership in [[Roman Egypt]], variation of the size of houses in societies from ancient Greece to Aztec Mexico, and inheritance and dowries in Babylonian society. Other data does not directly document variations in wealth or income but are known to reflect inequality, such as the ratio of rents to wages or of labor to capital.<ref>{{cite book|author=[[Walter Scheidel]]|title=The Great Leveler: Violence and the History of Inequality from the Stone Age to the Twenty-First Century|publisher=Princeton University Press|date=2017|isbn=978-0-691-16502-8|pages=15–16}}</ref>

== Other uses ==
Although the Gini coefficient is most popular in economics, it can, in theory, be applied in any field of science that studies a distribution. For example, in ecology, the Gini coefficient has been used as a measure of [[biodiversity]], where the cumulative proportion of species is plotted against the cumulative proportion of individuals.<ref name=natureArticle>{{cite journal | last1 = Wittebolle | first1 = Lieven | title = Initial community evenness favours functionality under selective stress | journal = [[Nature (journal)|Nature]] | year = 2009 | volume = 458 | issue = 7238 | pmid = 19270679 | pages = 623–626 | doi = 10.1038/nature07840| display-authors = 2 | last2 = Marzorati | first2 = Massimo | last3 = Balloi | first3 = Annalisa | last4 = Daffonchio | first4 = Daniele | last5 = Heylen | first5 = Kim | last6 = De Vos | first6 = Paul | last7 = Verstraete | first7 = Willy | last8 = Boon | first8 = Nico | bibcode = 2009Natur.458..623W | s2cid = 4419280 }}</ref> In health, it has been used as a measure of the inequality of health-related [[quality of life]] in a population.<ref name=popHealthArticle>{{cite journal | last=Asada | first=Yukiko | title = Assessment of the health of Americans: the average health-related quality of life and its inequality across individuals and groups | journal = Population Health Metrics | year = 2005 | volume = 3 | pmid=16014174 | page = 7 | pmc=1192818 | doi = 10.1186/1478-7954-3-7 | doi-access=free }}</ref> In education, it has been used as a measure of the inequality of universities.<ref name=MinervaArticle>{{cite journal | last1= Halffman | first1= Willem | last2= Leydesdorff | first2= Loet | title = Is Inequality Among Universities Increasing? Gini Coefficients and the Elusive Rise of Elite Universities | journal = Minerva | year = 2010 | volume = 48 | pmid= 20401157 | issue= 1 | pages = 55–72 | pmc= 2850525 | doi = 10.1007/s11024-010-9141-3| arxiv= 1001.2921 }}</ref> In chemistry it has been used to express the selectivity of [[protein kinase inhibitors]] against a panel of kinases.<ref name=JMedChemArticle>{{cite journal | last = Graczyk | first = Piotr | title = Gini Coefficient: A New Way To Express Selectivity of Kinase Inhibitors against a Family of Kinases | journal = Journal of Medicinal Chemistry | year = 2007 | volume = 50 | issue = 23 | pmid = 17948979 | pages = 5773–5779 | doi = 10.1021/jm070562u}}</ref> In engineering, it has been used to evaluate the fairness achieved by Internet routers in scheduling packet transmissions from different flows of traffic.<ref name=GreedyFairQueueing>{{Cite book |first1=Hongyuan |last1=Shi |first2=Harish |last2=Sethu |contribution=Greedy Fair Queueing: A Goal-Oriented Strategy for Fair Real-Time Packet Scheduling |pages=345–356 |title=Proceedings of the 24th IEEE Real-Time Systems Symposium |publisher=[[IEEE Computer Society]] |isbn=978-0-7695-2044-5 |year=2003}}</ref> In [[machine learning]], it has been used as a unified metric for evaluating many-versus-many (all-to-all) similarity in vector spaces across various data types, including images and text, and to show their effectiveness in guiding machine learning training sample selection, especially in sparse information settings.<ref>{{Citation | last1 = Fauber | first1 = Ben | title = Gini Coefficient as a Unified Metric for Evaluating Many-versus-Many Similarity in Vector Spaces. | date = 2024 | volume = abs/2411.07983 | arxiv = 2411.07983 }}</ref>

The Gini coefficient is sometimes used for the measurement of the discriminatory power of [[credit rating|rating]] systems in [[credit risk]] management.<ref>{{cite book|title=The Analytics of Risk Model Validation (Quantitative Finance)|editor1-first=George A.|editor1-last=Christodoulakis|editor2-first=Stephen|editor2-last=Satchell|isbn=978-0-7506-8158-2|date=November 2007|publisher=Academic Press}}</ref>

A 2005 study accessed US census data to measure home computer ownership and used the Gini coefficient to measure inequalities amongst whites and African Americans. Results indicated that although decreasing overall, home computer ownership inequality was substantially smaller among white households.<ref>{{cite journal|last1=Chakraborty|first1=J|last2=Bosman|first2=MM|title=Measuring the digital divide in the United States: race, income, and personal computer ownership|journal=Prof Geogr|year=2005|volume=57|issue=3|pages=395–410|doi=10.1111/j.0033-0124.2005.00486.x|bibcode=2005ProfG..57..395C|s2cid=154401826}}</ref>

A 2016 peer-reviewed study titled Employing the Gini coefficient to measure participation inequality in treatment-focused Digital Health Social Networks<ref>{{cite journal|last1=van Mierlo|first1=T|last2=Hyatt|first2=D|last3=Ching|first3=A|title=Employing the Gini coefficient to measure participation inequality in treatment-focused Digital Health Social Networks|journal=Netw Model Anal Health Inform Bioinforma|date=2016|volume=5|issue=32|pages=32|doi=10.1007/s13721-016-0140-7|pmid=27840788|pmc=5082574}}</ref> illustrated that the Gini coefficient was helpful and accurate in measuring shifts in inequality, however as a standalone metric it failed to incorporate overall network size.

Discriminatory power refers to a credit risk model's ability to differentiate between defaulting and non-defaulting clients. The formula <math>G_1</math>, in the calculation section above, may be used for the final model and at the individual model factor level to quantify the discriminatory power of individual factors. It is related to the accuracy ratio in population assessment models.

The Gini coefficient has also been applied to analyze inequality in [[Online dating application|dating apps]].<ref>{{Cite web|last=worst-online-dater|date=2015-03-25|title=Tinder Experiments II: Guys, unless you are really hot you are probably better off not wasting your…|url=https://medium.com/@worstonlinedater/tinder-experiments-ii-guys-unless-you-are-really-hot-you-are-probably-better-off-not-wasting-your-2ddf370a6e9a|access-date=2021-04-28|website=Medium|language=en}}</ref><ref>{{Cite web|last=Kopf|first=Dan|title=These statistics show why it's so hard to be an average man on dating apps|url=https://qz.com/1051462/these-statistics-show-why-its-so-hard-to-be-an-average-man-on-dating-apps/|access-date=2021-04-28|website=Quartz|date=15 August 2017 |language=en}}</ref>

Kaminskiy and Krivtsov<ref>{{cite book |last1= Kaminskiy |first1= M.P.|last2= Krivtsov|first2= V.V.|date= 2011|chapter= A Gini-Type Index for Aging/Rejuvenating Objects|title= Mathematical and Statistical Models and Methods in Reliability|location= Birkhäuser Boston|url = https://www.springer.com/gp/book/9780817649708| publisher= Springer|pages= 133–140|isbn=978-0-8176-4970-8}}</ref> extended the concept of the Gini coefficient from economics to [[reliability theory]] and proposed a Gini-type coefficient that helps to assess the degree of aging of non-repairable systems or aging and rejuvenation of repairable systems.  The coefficient is defined between −1 and 1 and can be used in both empirical and parametric life distributions. It takes negative values for the class of decreasing failure rate distributions and point processes with decreasing failure intensity rate and is positive for the increasing failure rate distributions and point processes with increasing failure intensity rate. The value of zero corresponds to the [[Exponential distribution|exponential life distribution]] or the [[Poisson point process#Homogeneous Poisson point process|Homogeneous Poisson Process]].

== See also ==
{{Columns-list|colwidth=22em|
* [[Diversity index]]
* [[Economic inequality]]
* [[Great Gatsby curve]]
* [[Herfindahl–Hirschman Index]]
* [[Hoover index]] (a.k.a. Robin Hood index)
* [[Human Poverty Index]]
* [[Income inequality metrics]]
* [[Kuznets curve]]
* [[List of countries by income equality]]
* [[List of countries by inequality-adjusted Human Development Index]]
* [[List of countries by wealth inequality]]
* [[List of U.S. states by Gini coefficient]]
* [[Lorenz curve]]
* [[Matthew effect]]
* [[Pareto distribution]]
* [[ROC analysis]]
* [[Suits index]]
* [[The Elephant Curve]]
* [[Utopia]]
* [[Welfare spending|Welfare]]
* [[Welfare economics]]
}}

== References ==
{{Reflist|30em}}

== Further reading ==
{{Refbegin|60em}}
* {{Cite book | last1=Amiel | first1=Y. | last2=Cowell | first2=F.&nbsp;A. | year=1999 | title=Thinking about Inequality | publisher=Cambridge | isbn=978-0-521-46696-7}}
* {{Cite book | first=Sudhir | last=Anand | year=1983 | title=Inequality and Poverty in Malaysia | publisher=Oxford University Press | location=New York | isbn=978-0-19-520153-6}}
* {{Cite journal | last=Brown | first=Malcolm | title=Using Gini-Style Indices to Evaluate the Spatial Patterns of Health Practitioners: Theoretical Considerations and an Application Based on Alberta Data | journal=Social Science & Medicine | year=1994 | volume=38 | pages=1243–1256 | doi=10.1016/0277-9536(94)90189-9 | pmid=8016689 | issue=9}}
* {{Cite book | first=S.&nbsp;R. | last=Chakravarty | year=1990 | title=Ethical Social Index Numbers | publisher=Springer-Verlag | location=New York | isbn=978-0-387-52274-6 }}
* {{Cite book | first=Angus | last=Deaton | year=1997 | title=Analysis of Household Surveys | publisher=Johns Hopkins University Press | location=Baltimore MD | isbn=978-0-585-23787-9 }}
* {{Cite journal | title=Bootstrapping the Gini coefficient of inequality | journal=Ecology | year=1987 | volume=68 | pages=1548–1551 | doi=10.2307/1939238 | jstor=1939238 | issue=5 |last1 = Dixon| first1=Philip M. | last2=Weiner | first2=Jacob  | last3=Mitchell-Olds | first3=Thomas | last4=Woodley | first4=Robert | bibcode=1987Ecol...68.1548D | s2cid=84940050 }}
* {{Cite journal | last=Dorfman | first=Robert | title=A Formula for the Gini Coefficient | journal=The Review of Economics and Statistics | year=1979 | volume=61 | pages=146–149 | doi=10.2307/1924845 | jstor=1924845 | issue=1 }}
* {{Cite book | first=Glenn | last=Firebaugh | year=2003 | title=The New Geography of Global Income Inequality | publisher=Harvard University Press | location=Cambridge, Massachusetts | isbn=978-0-674-01067-3 }}
* {{Cite journal | last=Gastwirth | first=Joseph L. | title=The Estimation of the Lorenz Curve and Gini Index | journal=The Review of Economics and Statistics | year=1972 | volume=54 | pages=306–316 | doi=10.2307/1937992 | jstor=1937992 | issue=3 }}
* {{Cite journal | last=Giles | first=David | title=Calculating a Standard Error for the Gini Coefficient: Some Further Results | journal=Oxford Bulletin of Economics and Statistics | year=2004 | volume=66 | issue=3 | pages=425–433 | doi=10.1111/j.1468-0084.2004.00086.x | url=http://web.uvic.ca/econ/ewp0202.pdf | citeseerx=10.1.1.202.6462 | s2cid=16972099 | archive-url=https://web.archive.org/web/20040505124219/http://web.uvic.ca/econ/ewp0202.pdf | archive-date=5 May 2004 }}
* {{Cite book | last=Gini | first=Corrado | year=1912 | title=Variabilità e mutabilità | bibcode=1912vamu.book.....G }} Reprinted in {{cite book | title=Memorie di metodologica statistica | editor1-last=Pizetti | editor1-first=E. | editor2-last=Salvemini | editor2-first=T. | location=Rome | publisher=Libreria Eredi Virgilio Veschi | year=1955}}
* {{Cite journal | last=Gini | first=Corrado | title=Measurement of Inequality of Incomes | journal=The Economic Journal | year=1921 | volume=31 | pages=124–126 | doi=10.2307/2223319 | jstor=2223319 | issue=121 | url=https://zenodo.org/record/1449432 }}
* {{cite journal | last=Giorgi | first=Giovanni Maria | year=1990 | title=Bibliographic portrait of the Gini concentration ratio | journal=Metron | volume=48 | pages=183–231 | url=http://econwpa.repec.org/eps/em/papers/0511/0511004.pdf | archive-url=https://web.archive.org/web/20160804055002/http://econwpa.repec.org/eps/em/papers/0511/0511004.pdf | archive-date=4 August 2016 }}
* {{Cite journal | last1=Karagiannis | first1=E.  | last2=Kovacevic | first2=M. | title=A Method to Calculate the Jackknife Variance Estimator for the Gini Coefficient | journal=Oxford Bulletin of Economics and Statistics | year=2000 | volume=62 | pages=119–122 | doi=10.1111/1468-0084.00163}}
* {{Cite journal | last1=Mills | first1=Jeffrey A. | last2=Zandvakili | first2=Sourushe | title=Statistical Inference via Bootstrapping for Measures of Inequality | journal=Journal of Applied Econometrics | year=1997 | volume=12 | issue=2 | pages=133–150 | doi=10.1002/(SICI)1099-1255(199703)12:2<133::AID-JAE433>3.0.CO;2-H | jstor=2284908 | url=http://www.levyinstitute.org/pubs/wp136.pdf |archive-url=https://web.archive.org/web/20120718000000/http://www.levyinstitute.org/pubs/wp136.pdf |archive-date=2012-07-18 |url-status=live | citeseerx=10.1.1.172.5003 | hdl=10419/186818 }}
* {{Cite journal | last1=Modarres | first1=Reza | last2=Gastwirth | first2=Joseph L. | title=A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality | journal=Oxford Bulletin of Economics and Statistics | year=2006 | volume=68 | issue=3 | pages=385–390 | doi=10.1111/j.1468-0084.2006.00167.x| s2cid=122716409 }}
* {{Cite journal | last=Morgan | first=James | title=The Anatomy of Income Distribution | journal=The Review of Economics and Statistics | year=1962 | volume=44 | pages=270–283 | doi=10.2307/1926398 | jstor=1926398 | issue=3 }}
* {{Cite journal | last=Ogwang | first=Tomson | title=A Convenient Method of Computing the Gini Index and its Standard Error | journal=Oxford Bulletin of Economics and Statistics | year=2000 | volume=62 | pages=123–129 | doi=10.1111/1468-0084.00164 }}
* {{Cite journal | last=Ogwang | first=Tomson | title=Calculating a Standard Error for the Gini Coefficient: Some Further Results: Reply | journal=Oxford Bulletin of Economics and Statistics | year=2004 | volume=66 | issue=3 | pages=435–437 | doi=10.1111/j.1468-0084.2004.00087.x| s2cid=122160535 }}
* {{cite journal |last1=Xu |first1=Kuan |title=How has the Literature on Gini's Index Evolved in the Past 80 Years? |date=January 2004 | url=http://economics.dal.ca/RePEc/dal/wparch/howgini.pdf | archive-url=https://web.archive.org/web/20060928082244/http://economics.dal.ca/RePEc/dal/wparch/howgini.pdf | access-date=1 June 2006 | archive-date=28 September 2006 |journal=[[Social Science Research Network|SSRN Electronic Journal]] |doi=10.2139/ssrn.423200 |series=Economics Working Paper |at=[[Dalhousie University]]}} The Chinese version of this paper was published as {{cite journal | last=Xu | first=Kuan | title=How Has the Literature on Gini's Index Evolved in the Past 80 Years? | journal = China Economic Quarterly | year=2003 | volume=2 | pages=757–778 }}
* {{Cite journal | last=Yitzhaki | first=Shlomo  | title=Calculating Jackknife Variance Estimators for Parameters of the Gini Method | journal=Journal of Business and Economic Statistics | year=1991 | volume=9 | pages=235–239 | doi=10.2307/1391792 | jstor=1391792 | issue=2 }}
{{Refend}}

== External links ==
{{Commons category}}
* Deutsche Bundesbank: [https://web.archive.org/web/20081218111312/http://www.bundesbank.de/download/bankenaufsicht/dkp/200503dkp_b.pdf Do banks diversify loan portfolios?], 2005 (on using e.g. the Gini coefficient for risk evaluation of loan portfolios)
* [https://www.forbes.com/forbes/2003/0317/098.html Forbes: In praise of inequality]
* [http://www.theresearchkitchen.com/archives/219 Measuring Software Project Risk With The Gini Coefficient], an application of the Gini coefficient to software
* [http://web.worldbank.org/WBSITE/EXTERNAL/TOPICS/EXTPOVERTY/EXTPA/0,,contentMDK:20238991~menuPK:492138~pagePK:148956~piPK:216618~theSitePK:430367,00.html The World Bank: Measuring Inequality]
* [http://utip.gov.utexas.edu/tutorials/theo_basic_ineq_measures.doc Travis Hale, University of Texas Inequality Project:The Theoretical Basics of Popular Inequality Measures], online computation of examples: [http://www.poorcity.richcity.org/calculator/?quantiles=7,*18000|10,*22000|280,*25000|15,*35000|15,*40000|50,*60000|10,*75000|6,*80000|4,*120000|2,*200000|1,1000000 1A], [http://www.poorcity.richcity.org/calculator/?quantiles=12,*15000|25,*20000|1000,*30000|35,*35000|100,*45000|80,*50000|10,*60000|25,*80000|8,*175000|4,*250000|1,5000000 1B]
* [http://image.guardian.co.uk/sys-files/Guardian/documents/2009/03/13/inequality.pdf Article from The Guardian analysing inequality in the UK 1974–2006]
* [https://www.wider.unu.edu/project/wiid-world-income-inequality-database World Income Inequality Database]
* [https://www.bbc.com/news/blogs-magazine-monitor-31847943 BBC News: What is the Gini coefficient?]
* [http://www.oecd.org/els/social/inequality/GU Income Distribution and Poverty in OECD Countries]
* [http://inequality.org/unequal-americas-income-distribution/ U.S. Income Distribution: Just How Unequal?]

{{Globalization}}
{{Deprivation Indicators}}
{{Poverty}}
{{Authority control}}

[[Category:1912 introductions]]
[[Category:1912 in economic history]]
[[Category:Concentration indicators]]
[[Category:Demographic economics]]
[[Category:Income inequality metrics]]
[[Category:Welfare economics]]
[[Category:Italian inventions]]