# Gini coefficient

The Gini coefficient (sometimes expressed as a Gini ratio or a normalized Gini index) (/ini/ jee-nee) is a measure of statistical dispersion intended to represent the income or wealth distribution of a nation's residents, and is the most commonly used measure of inequality. It was developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper Variability and Mutability (Italian: Variabilità e mutabilità).

The Gini coefficient measures the inequality among values of a frequency distribution (for example, levels of income). A Gini coefficient of zero expresses perfect equality, where all values are the same (for example, where everyone has the same income). A Gini coefficient of 1 (or 100%) expresses maximal inequality among values (e.g., for a large number of people, where only one person has all the income or consumption, and all others have none, the Gini coefficient will be very nearly one). However, a value greater than one may occur if some persons represent negative contribution to the total (for example, having negative income or wealth). For larger groups, values close to or above 1 are very unlikely in practice. Given the normalization of both the cumulative population and the cumulative share of income used to calculate the Gini coefficient, the measure is not overly sensitive to the specifics of the income distribution, but rather only on how incomes vary relative to the other members of a population. The exception to this is in the redistribution of wealth resulting in a minimum income for all people. When the population is sorted, if their income distribution were to approximate a well known function, then some representative values could be calculated.

Income Distribution function PDF(x) Gini Coefficient
Dirac delta function ${\displaystyle \delta (x-x_{0}),\,x_{0}>0}$ 0
Uniform distribution ${\displaystyle {\begin{cases}{\frac {1}{b-a}}&a\leq x\leq b\\0&\mathrm {otherwise} \end{cases}}}$ ${\displaystyle {\frac {b-a}{3(b+a)}}}$
Exponential distribution ${\displaystyle \lambda e^{-x\lambda },\,\,x>0}$ ${\displaystyle 1/2}$
Log-normal distribution ${\displaystyle {\frac {1}{\sigma {\sqrt {2\pi }}}}e^{\frac {-(\ln \,(x)-\mu )^{2}}{\sigma ^{2}}}}$ ${\displaystyle {\textrm {erf}}(\sigma /2)}$
Pareto distribution ${\displaystyle {\begin{cases}{\frac {\alpha k^{\alpha }}{x^{\alpha +1}}}&x\geq k\\0&x ${\displaystyle {\begin{cases}1&0<\alpha <1\\{\frac {1}{2\alpha -1}}&\alpha \geq 1\end{cases}}}$
Chi-squared distribution ${\displaystyle {\frac {2^{-k/2}e^{-x/2}x^{k/2-1}}{\Gamma (k/2)}}}$ ${\displaystyle {\frac {2\,\Gamma \left({\frac {1+k}{2}}\right)}{k\,\Gamma (k/2){\sqrt {\pi }}}}}$
Gamma distribution ${\displaystyle {\frac {e^{-x/\theta }x^{k-1}\theta ^{-k}}{\Gamma (k)}}}$ ${\displaystyle {\frac {\Gamma \left({\frac {2k+1}{2}}\right)}{k\,\Gamma (k){\sqrt {\pi }}}}}$
Weibull distribution ${\displaystyle {\frac {k}{\lambda }}\,\left({\frac {x}{\lambda }}\right)^{k-1}e^{-(x/\lambda )^{k}}}$ ${\displaystyle 1-2^{-1/k}}$
Beta distribution ${\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{B(\alpha ,\beta )}}}$ ${\displaystyle \left({\frac {2}{\alpha }}\right){\frac {B(\alpha +\beta ,\alpha +\beta )}{B(\alpha ,\alpha )B(\beta ,\beta )}}}$
Income Gini coefficient
World, 1820–2005
Year World Gini coefficients
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
Year World Gini coefficient
1988 .80
1993 .76
1998 .74
2003 .72
2008 .70
2013 .65
Table A. Different income distributions
with the same Gini Index
Household
Group
Country A
Annual
Income ($) Country B Annual Income ($)
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$200,000
Country's Gini 0.2 0.2
Table B. Same income distributions
but different Gini Index
Household
number
Country A
Annual
Income ($) Household combined number Country A combined Annual Income ($)
1 20,000 1 & 2 50,000
2 30,000
3 40,000 3 & 4 90,000
4 50,000
5 60,000 5 & 6 130,000
6 70,000
7 80,000 7 & 8 170,000
8 90,000
9 120,000 9 & 10 270,000
10 150,000
Total Income $710,000$710,000
Country's Gini 0.303 0.293
Table C. Household money income
distributions and Gini Index, USA
Income bracket
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%