Gini coefficient
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Graphical representation of the Gini coefficient

The Gini coefficient is a measure of inequality of a distribution. It is defined as a ratio with values between 0 and 1: the numerator is the area between the Lorenz curve of the distribution and the uniform distribution line; the denominator is the area under the uniform distribution line. It was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability"). The Gini index is the Gini coefficient expressed as a percentage, and is equal to the Gini coefficient multiplied by 100. (The Gini coefficient is equal to half of the relative mean difference.) The Gini coefficient is often used to measure income inequality. Here, 0 corresponds to perfect income equality (i.e. everyone has the same income) and 1 corresponds to perfect income inequality (i.e. one person has all the income, while everyone else has zero income). The Gini coefficient can also be used to measure wealth inequality. This use requires that no one has a negative net wealth. It is also commonly used for the measurement of discriminatory power of rating systems in the credit risk management.

Calculation

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini coefficient is A/(A+B). Since A+B = 0.5, the Gini coefficient, G = 2A = 1-2B. If the Lorenz curve is represented by the function Y = L(X), the value of B can be found with integration and:

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:
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For a population with values yi, i = 1 to n, that are indexed in non-decreasing order ( yi ≤ yi+1):

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For a discrete probability function f(y), where yi, i = 1 to n, are the points with nonzero probabilities and which are indexed in increasing order ( yi < yi+1):

where: and
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For a cumulative distribution function F(y) that is piecewise differentiable, has a mean μ, and is zero for all negative values of y:

Since the Gini coefficient is half the relative mean difference, it can also be calculated using formulas for the relative mean difference. For a random sample S consisting of values yi, i = 1 to n, that are indexed in non-decreasing order ( yi ≤ yi+1), the statistic:

is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like the relative mean difference, there does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient. Confidence intervals for the population Gini coefficient can be calculated using bootstrap techniques. 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 by using various techniques for interpolating the missing values of the Lorenz curve. If ( X k , Yk ) are the known points on the Lorenz curve, with the X k indexed in increasing order ( X k - 1 < X k ), so that:
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Xk is the cumulated proportion of the population variable, for k = 0,...,n, with X0 = 0, Xn = 1. Yk is the cumulated proportion of the income variable, for k = 0,...,n, with Y0 = 0, Yn = 1.

If the Lorenz curve is approximated on each interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

is the resulting approximation for G. More accurate results can be obtained using other methods to approximate the area B, such as approximating the Lorenz curve with a 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.

While most developed European nations tend to have Gini coefficients between 0.24 and 0.36, the United States Gini coefficient is above 0.4, indicating that the United States has greater inequality. Using the Gini can help quantify differences in welfare and 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 (see criticisms section).

Correlation with per-capita GDP
Poor countries (those with low per-capita GDP) have Gini coefficients that fall over the whole range from low (0.25) to high (0.71), while rich countries have generally low Gini coefficient (under 0.40).

Advantages as a measure of inequality
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The Gini coefficient's main advantage is that it is a measure of inequality by means of a ratio analysis, rather than a variable unrepresentative of most of the population, such as per capita income or gross domestic product. It can be used to compare income distributions across different population sectors as well as countries, for example the Gini coefficient for urban areas differs from that of rural areas in many countries (though the United States' urban and rural Gini coefficients are nearly identical). It is sufficiently simple that it can be compared across countries and be easily interpreted. GDP statistics are often criticised as they do not represent changes for the whole population; the Gini coefficient demonstrates how income has changed for poor and rich. If the Gini coefficient is rising as well as GDP, poverty may not be improving for the majority of the population. The Gini coefficient can be used to indicate how the distribution of income has changed within a country over a period of time, thus it is possible to see if inequality is increasing or decreasing. The Gini coefficient satisfies four important principles: o Anonymity: it does not matter who the high and low earners are.

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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. o Population independence: it does not matter how large the population of the country is. o Transfer principle: if income (less than the difference), is transferred from a rich person to a poor person the resulting distribution is more equal. o Disadvantages as a measure of inequality
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The Gini coefficient measured for a large economically diverse country will generally result in a much higher coefficient than each of its regions has individually. For this reason the scores calculated for individual countries within the EU are difficult to compare with the score of the entire US. Comparing income distributions among countries may be difficult because benefits systems may differ. For example, some countries give benefits in the form of money while others give food stamps, which may not be counted as income in the Lorenz curve and therefore not taken into account in the Gini coefficient. The measure will give different results when applied to individuals instead of households. When different populations are not measured with consistent definitions, comparison is not meaningful. The Lorenz curve may understate the actual amount of inequality if richer households are able to use income more efficiently than lower income households. From another point of view, measured inequality may be the result of more or less efficient use of household incomes. As for all statistics, there will be systematic and random errors in the data. The meaning of the Gini coefficient decreases as the data become less accurate. Also, countries may collect data differently, making it difficult to compare statistics between countries. Economies with similar incomes and Gini coefficients can still have very different income distributions. This is because the Lorenz curves can have different shapes and yet still yield the same Gini coefficient. As an extreme

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example, an economy where half the households have no income, and the other half share income equally has a Gini coefficient of ½; but an economy with complete income equality, except for one wealthy household that has half the total income, also has a Gini coefficient of ½.
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Too often only the Gini coefficient is quoted without describing the proportions of the quantiles used for measurement. As with other inequality coefficients, the Gini coefficient is influenced by the granularity of the measurements. For example, five 20% quantiles (low granularity) will yield a lower Gini coefficient than twenty 5% quantiles (high granularity) taken from the same distribution.

As one result of this criticism, additionally to or in competition with the Gini coefficient entropy measures are frequently used (e.g. the Atkinson and Theil indices). These measures attempt to compare the distribution of resources by intelligent players in the market with a maximum entropy random distribution, which would occur if these players acted like non-intelligent particles in a closed system following the laws of statistical physics.

A lower Gini coefficient tends to indicate a higher level of social and economic equality. Gini Richest 10% to poorest index 10% 19 3.3 24.7 8.1 24.9 4.5 25 6.2 25.4 5.2 25.8 6.1 25.8 6.7 26.2 5.4 26.8 6.1 26.9 5.5 26.9 5.6 28.1 5.9 28.2 5.9 28.3 6.9 28.4 5.9 28.9 5.8 29 7.3 29.1 6.9 29.2 7 29.7 6.9 30 6.6 30.3 6.4 30.3 17.8 30.6 6.5 30.9 9.2 31 7.5 31.6 7.8 31.8 6.8 32.5 7.3 32.6 7.8 32.6 9.4 32.7 9.1 33 8.2 33.2 8.1 33.2 8.2 Richest 20% to poorest 20% 2.6 4.3 3.4 4 3.5 3.9 4 3.8 4 3.8 3.8 4.1 4.1 4.3 3.9 4 4.8 4.4 4.4 4.5 4.3 4.4 9.1 4.3 5.1 4.9 4.7 4.6 4.9 5.2 5.5 5.6 4.9 5.1 5.3 Survey year 2002 1997 1993 2000 1996 2000 1996 2001 2000 2002 2000 2003 2002 2000 1998–99 1983–85 2001 2000 2003 2002 1999–00 2003 1998 2002 1999 2003 1998 2000 1999–00 2003 2000 1995 2000 1999–00 2003

Rank 1 2 3 4 5 6 6 8 9 10 10 12 13 14 15 16 17 18 19 20 21 22 22 24 25 26 27 28 29 30 30 32 33 34 34

Country Azerbaijan Denmark Japan Sweden Czech Republic Norway Slovakia Bosnia and Herzegovina Uzbekistan Hungary Finland Ukraine Albania Germany Slovenia Rwanda Croatia Austria Bulgaria Belarus Ethiopia Kyrgyzstan Mongolia Pakistan Netherlands Romania South Korea Bangladesh India Tajikistan Canada France Belgium Sri Lanka Moldova

36 37 38 39 40 40 40 43 44 45 45 47 48 49 50 51 51 51 54 55 56 57 58 59 60 61 61 63 64 64 66 67 68 69 69 71 71 73 73

Yemen Switzerland Armenia Kazakhstan Indonesia Ireland Greece Egypt Poland Tanzania Laos Spain Australia Algeria Estonia Lithuania Italy United Kingdom New Zealand Benin Vietnam Latvia Jamaica Portugal Jordan Republic of Macedonia Mauritania Israel Morocco Burkina Faso Mozambique Tunisia Russia Guinea Trinidad and Tobago Georgia Cambodia Ghana United States

33.4 33.7 33.8 33.9 34.3 34.3 34.3 34.4 34.5 34.6 34.6 34.7 35.2 35.3 35.8 36 36 36 36.2 36.5 37 37.7 37.9 38.5 38.8 39 39 39.2 39.5 39.5 39.6 39.8 39.9 40.3 40.3 40.4 40.4 40.8 40.8

8.6 9 8 8.5 7.8 9.4 10.2 8 8.8 9.2 8.3 10.3 12.5 9.6 10.8 10.4 11.6 13.8 12.5 9.4 9.4 11.6 11.4 15 11.3 12.5 12 13.4 11.7 11.6 12.5 13.4 12.7 12.3 14.4 15.4 11.6 14.1 15.9

5.6 5.5 5 5.6 5.2 5.6 6.2 5.1 5.6 5.8 5.4 6 7 6.1 6.4 6.3 6.5 7.2 6.8 6 6 6.8 6.9 8 6.9 7.5 7.4 7.9 7.2 6.9 7.2 7.9 7.6 7.3 8.3 8.3 6.9 8.4 8.4

1998 2000 2003 2003 2002 2000 2000 1999–00 2002 2000–01 2002 2000 1994 1995 2003 2003 2000 1999 1997 2003 2002 2003 2000 1997 2002–03 2003 2000 2001 1998–99 2003 1996–97 2000 2002 1994 1992 2003 1997 1998–99 2000

73 76 77 78 79 80 80 82 82 84 85 86 87 87 89 90 90 92 93 94 95 96 97 98 99 100 101 102 103 104 104 106 107 108 109 110 111 112 113

Turkmenistan Senegal Thailand Zambia Burundi Singapore Kenya Uganda Iran Nicaragua Hong Kong, China (SAR) Turkey Nigeria Ecuador Venezuela Côte d’Ivoire Cameroon People's Republic of China Uruguay Philippines Guinea-Bissau Nepal Madagascar Malaysia Mexico Costa Rica Zimbabwe Gambia Malawi Niger Mali Papua New Guinea Dominican Republic El Salvador Argentina Honduras Peru Guatemala Panama

40.8 41.3 42 42.1 42.4 42.5 42.5 43 43 43.1 43.4 43.6 43.7 43.7 44.1 44.6 44.6 44.7 44.9 46.1 47 47.2 47.5 49.2 49.5 49.9 50.1 50.2 50.3 50.5 50.5 50.9 51.7 52.4 52.8 53.8 54.6 55.1 56.4

12.3 12.8 12.6 13.9 19.3 17.7 13.6 14.9 17.2 15.5 17.8 16.8 17.8 44.9 20.4 16.6 15.7 18.4 17.9 16.5 19 15.8 19.2 22.1 24.6 30 22 20.2 22.7 46 23.1 23.8 30 57.5 34.5 34.2 40.5 48.2 54.7

7.7 7.5 7.7 8 9.5 9.7 8.2 8.4 9.7 8.8 9.7 9.3 9.7 17.3 10.6 9.7 9.1 10.7 10.2 9.7 10.3 9.1 11 12.4 12.8 14.2 12 11.2 11.6 20.7 12.2 12.6 14.4 20.9 17.6 17.2 18.6 20.3 23.9

1998 1995 2002 2002–03 1998 1998 1997 1999 1998 2001 1996 2003 2003 1998 2000 2002 2001 2001 2003 2000 1993 2003–04 2001 1997 2002 2001 1995 1998 1997 1995 1994 1996 2003 2002 2003 2003 2002 2002 2002

114 115 115 117 118 119 120 121 122 123 124 125 126

Chile Paraguay South Africa Brazil Colombia Haiti Bolivia Swaziland Central African Republic Sierra Leone Botswana Lesotho Namibia

57.1 57.8 57.8 58 58.6 59.2 60.1 60.9 61.3 62.9 63 63.2 74.3

40.6 73.4 33.1 57.8 63.8 71.7 168.1 49.7 69.2 87.2 77.6 105 128.8

18.7 27.8 17.9 23.7 25.3 26.6 42.3 23.8 32.7 57.6 31.5 44.2 56.1

2000 2002 2000 2003 2003 2001 2002 1994 1993 1989 1993 1995 1993

United Nations 2006 Development Programme Report (p. 335).

year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

China’s Gini Coefficient 0.38 0.4 0.4 0.41 0.41 0.41 0.41 0.41 0.42 0.46 0.45 0.447 0.447 0.447

Source: Ravallion and Chen, 2004. China Statistical Yearbook (State Statistical Bureau, 1992 1996 and 1997
2001). http://www3.nccu.edu.tw/~jthuang/inequality.pdf

Gini Coefficient for China’s Income Distribution, 1981-2003

Source: Ravallion and Chen, Measuring Pro-Poor Growth, World Bank Policy Research Working Paper 2666. August, 2001; The World Bank: Biannual on China’s Economy. Business Weekly: No. 9, 2004 http://www.cdrf.org.cn/2006cdf/news_Harmony.htm