# Gini coefficient

File:Economics Gini coefficient.svg
Graphical representation of the Gini coefficient
(The area of the whole triangle is defined as 1.)

The Gini coefficient is a measure of statistical dispersion most prominently used as a measure of inequality of income distribution or inequality of wealth 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. Thus, a low Gini coefficient indicates more equal income or wealth distribution, while a high Gini coefficient indicates more unequal distribution. 0 corresponds to perfect equality (everyone having exactly the same income) and 1 corresponds to perfect inequality (where one person has all the income, while everyone else has zero income). The Gini coefficient requires that no one have a negative net income or wealth.

The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability").

The Gini coefficient is also commonly used for the measurement of the discriminatory power of rating systems in credit risk management.

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.)

## 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 = A/(.5) = 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:

${\displaystyle G=1-2\,\int _{0}^{1}L(X)dX}$

In some cases, this equation can be applied to calculate the Gini coefficient without direct reference to the Lorenz curve. For example:

• For a population uniform on the values yi, i = 1 to n, indexed in non-decreasing order ( yiyi+1):
${\displaystyle G={\frac {1}{n}}\left(n+1-2\left({\frac {\Sigma _{i=1}^{n}\;(n+1-i)y_{i}}{\Sigma _{i=1}^{n}y_{i}}}\right)\right)}$
• 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):
${\displaystyle G=1-{\frac {\Sigma _{i=1}^{n}\;f(y_{i})(S_{i-1}+S_{i})}{S_{n}}}}$
where:
${\displaystyle S_{i}=\Sigma _{j=1}^{i}\;f(y_{j})\,y_{j}\,}$ and ${\displaystyle S_{0}=0\,}$
${\displaystyle G=1-{\frac {1}{\mu }}\int _{0}^{\infty }(1-F(y))^{2}dy}$

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 ( yiyi+1), the statistic:

${\displaystyle G(S)={\frac {1}{n-1}}\left(n+1-2\left({\frac {\Sigma _{i=1}^{n}\;(n+1-i)y_{i}}{\Sigma _{i=1}^{n}y_{i}}}\right)\right)}$

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:

• 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:

${\displaystyle G_{1}=1-\sum _{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1})}$

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.

## Income Gini coefficients in the world

A complete listing is in list of countries by income equality; the article economic inequality discusses the social and policy aspects of income and asset inequality.

File:World Map Gini coefficient.svg
Gini coefficient, income distribution by country.
 < 0.25   0.25–0.29 0.30–0.34   0.35–0.39   0.40–0.44 0.45–0.49   0.50–0.54   0.55–0.59   ≥ 0.60 N/A

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).

The Gini coefficient for the entire world has been estimated by various parties to be between 0.56 and 0.66.[1][2]

File:Gini since WWII.gif
Gini coefficients, income distribution over time for selected countries

### 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 intermediate Gini coefficient (under 0.40). Generally, the lowest Gini coefficients can be found in the Scandinavian countries, in the recently ex-socialist countries of Eastern Europe and in Japan.

### US income gini coefficients over time

Gini coefficients for the United States at various times, according to the US Census Bureau:

• 1967: 0.397 (first year reported)
• 1968: 0.386 (lowest coefficient reported)
• 1970: 0.394
• 1980: 0.403
• 1990: 0.428
• 2000: 0.462
• 2005: 0.469 (most recent year reported; highest coefficient reported)[3]

Between 1968 and 2005, the Gini coefficient fell in only seven years. Some argue this rise corresponds to the lowering of the highest tax bracket, for example, from 70% in the 1960s to 35% by 2000. However, many other variables that could affect the Gini coefficient have changed during this period as well. For example, much technological progress has occurred, eliminating formerly middle-class factory jobs in favor of the service sector; additionally, the economy has shifted towards professions that require higher education.

## Advantages of Gini coefficient as a measure of inequality

• 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:
• 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.

## Disadvantages of Gini coefficient as a measure of inequality

• The Gini coefficient of different sets of people cannot be averaged to obtain the Gini coefficient of all the people in the sets: if a Gini coefficient were to be calculated for each person it would always be zero. When measuring its value for a large, economically diverse country, a much higher coefficient than each of its regions has individually will result.
For this reason the scores calculated for individual countries within the EU are difficult to compare with the score of the entire US: the overall value for the EU should be used in that case, 31.3[4], which is still much lower than the United States', 45[5]. Using decomposable inequality measures (e.g. the Theil index ${\displaystyle T}$ converted by ${\displaystyle 1-{e^{-T}}}$ into a inequality coefficient) averts such problems.
• 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.
• 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 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. In practice, such distributions do not exist, and therefore, the impact of different but realistic curves is less obvious.

## Problems in using the Gini coefficient

• Gini coefficients do include income gained from wealth; however, the Gini coefficient is used to measure net income more than net worth, which can be misinterpreted. For example, Sweden has a low Gini coefficient for income distribution but a high Gini coefficient for wealth (5% of Swedish household shareholders hold 77% of the share value owned by households)[6]. In other words and as a normative statement: The Gini coefficient should be interpreted as measuring effective egalitarianism; and distribution of stock ownership does not appear to correlate to many recognized indicators of egalitarianism.
• 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 usually yield a lower Gini coefficient than twenty 5% quantiles (high granularity) taken from the same distribution. This is an often encountered problem with measurements.
• Care should be taken in using the Gini coefficient as a measure of egalitarianism, as it is properly a measure of income dispersion. Two equally egalitarian countries with different immigration policies may have different Gini coefficients.

## General problems of measurement

• 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[citation needed] 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.
• As for all statistics, there may 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.

As one result of this criticism, in addition 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 agents in the market with a maximum entropy random distribution, which would occur if these agents acted like non-intelligent particles in a closed system following the laws of statistical physics.

## Notes

1. [1]
2. [2]
3. Note that the calculation of the index for the United States was changed in 1992, resulting in an upwards shift of about 0.02 in the coefficient. Comparisons before and after that period may be misleading. (Data from the US Census Bureau.)
4. [https://www.cia.gov/library/publications/the-world-factbook/geos/ee.html CIA World Factbook—The European Union
5. CIA World Factbook—The United States
6. (Data from the Statistics Sweden.)

## References

• Amiel, Y.; Cowell, F.A. (1999). Thinking about Inequality. Cambridge.
• Anand, Sudhir (1983). Inequality and Poverty in Malaysia. New York: Oxford University Press.
• Brown, Malcolm (1994). "Using Gini-Style Indices to Evaluate the Spatial Patterns of Health Practitioners: Theoretical Considerations and an Application Based on Alberta Data". Social Science Medicine. 38: 1243–1256.
• Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: Springer-Verlag.
• Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987). "Bootstrapping the Gini coefficient of inequality". Ecology. 68: 1548–1551.
• Dorfman, Robert (1979). "A Formula for the Gini Coefficient". The Review of Economics and Statistics. 61: 146–149.
• Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index". The Review of Economics and Statistics. 54: 306–316.
• Gini, Corrado (1912). "Variabilità e mutabilità" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).
• Gini, Corrado (1921). "Measurement of Inequality and Incomes". The Economic Journal. 31: 124–126.
• Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics. 12: 133–150.
• Morgan, James (1962). "The Anatomy of Income Distribution". The Review of Economics and Statistics. 44: 270–283.
• Xu, Kuan (January, 2004). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". Department of Economics, Dalhousie University. Retrieved on 2006-06-01. The Chinese version of this paper appears in Xu, Kuan (2003). "How Has the Literature on Gini's Index Evolved in the Past 80 Years?". China Economic Quarterly. 2: 757–778.