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Gini Coefficient
Gini Coefficient

The Gini coefficient (sometimes expressed as a Gini ratio or a normalized Gini index) (/dʒini/ jeenee) 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 0 Uniform distribution Exponential distribution Lognormal distribution Pareto distribution Chisquared distribution Gamma distribution Weibull distribution Beta distribution Income Gini coefficient
World, 1820–2005Year 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 IndexHousehold
GroupCountry 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 $200,000 Country's Gini 0.2 0.2 Table B. Same income distributions
but different Gini IndexHousehold
numberCountry A
Annual
Income ($)Household
combined
numberCountry 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, USAIncome bracket
(in 2010 adjusted dollars)% of Population
1979% of Population
2010Under $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 pretax basis0.404 0.469
 This may be simplified to:
 This formula actually applies to any real population, since each person can be assigned his or her own y_{i}.
 is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like G, G (S) has a simpler form:
 .
 where
 If the points with nonzero probabilities are indexed in increasing order (y_{i} < y_{i+1}) then:
 where
 and . These formulae are also applicable in the limit as .

Income Distribution function PDF(x) Gini Coefficient Dirac delta function 0 Uniform distribution Exponential distribution Lognormal distribution Pareto distribution Chisquared distribution Gamma distribution Weibull distribution Beta distribution
 Different income distributions with the same Gini coefficient
 Extreme wealth inequality, yet low income Gini coefficient
 Small sample bias – sparsely populated regions more likely to have low Gini coefficient
 Gini coefficient is unable to discern the effects of structural changes in populations
 Inability to value benefits and income from informal economy affects Gini coefficient accuracy
 For a population uniform on the values y_{i}, i = 1 to n, indexed in nondecreasing order (y_{i} ≤ y_{i+1}):
 X_{k} is the cumulated proportion of the population variable, for k = 0,...,n, with X_{0} = 0, X_{n} = 1.
 Y_{k} is the cumulated proportion of the income variable, for k = 0,...,n, with Y_{0} = 0, Y_{n} = 1.
 Y_{k} should be indexed in nondecreasing order (Y_{k} > Y_{k – 1})
 Anonymity: it does not matter who the high and low earners are.
 Scale independence: the Gini coefficient does not consider the size of the economy, the way it is measured, or whether it is a rich or poor country on average.
 Population independence: it does not matter how large the population of the country is.
 Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal.
 Diversity index
 Economic inequality
 Great Gatsby curve
 Human Poverty Index
 Income inequality metrics
 Kuznets curve
 Pareto distribution
 Hoover index (a.k.a. Robin Hood index)
 ROC analysis
 Social welfare provision
 Suits index
 Utopia
 Welfare economics
 List of countries by distribution of wealth
 List of countries by income equality
 List of U.S. states by income equality
 Herfindahl index
 Deutsche Bundesbank: Do banks diversify loan portfolios?, 2005 (on using e.g. the Gini coefficient for risk evaluation of loan portfolios)
 Forbes Article, In praise of inequality
 Measuring Software Project Risk With The Gini Coefficient, an application of the Gini coefficient to software
 The World Bank: Measuring Inequality
 Travis Hale, University of Texas Inequality Project:The Theoretical Basics of Popular Inequality Measures, online computation of examples: 1A, 1B
 Article from The Guardian analysing inequality in the UK 1974–2006
 World Income Inequality Database
 Income Distribution and Poverty in OECD Countries
 U.S. Income Distribution: Just How Unequal?

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Gini Coefficient