Gini coefficient

The Gini coefficient (also known as the Gini index or Gini ratio) is a measure of statistical dispersion developed by the Italian statistician and sociologist Corrado Gini and published in his 1912 paper "Variability and Mutability" (Italian: Variabilità e mutabilità).[1][2]

The Gini coefficient is a measure of the inequality of a distribution, a value of 0 expressing perfect equality where everyone has equal shares of income and a value of 1 expresses maximal inequality where only one person has all the income.[3] It has found application in the study of inequalities in disciplines as diverse as sociology, economics, health science, ecology, chemistry, engineering and agriculture.[4]

It is commonly used as a measure of inequality of income or wealth.[5] Worldwide, Gini coefficients for income range from approximately 0.23 (Sweden) to 0.70 (Namibia) although not every country has been assessed.

Contents

Definition

The Gini coefficient is usually defined 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). 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., G=A/(A+B).

The Gini coefficient can range from 0 to 1; it is sometimes expressed as a percentage ranging between 0 and 100. More specifically, the upper bound of the Gini coefficient equals 1 only in populations of infinite size. In a population of size N, the upper bound is equal to 1 − 2 / (N + 1).

A low Gini coefficient indicates a more equal distribution, with 0 corresponding to complete equality, while higher Gini coefficients indicate more unequal distribution, with 1 corresponding to complete inequality. To be validly computed, no negative goods can be distributed. Thus, if the Gini coefficient is being used to describe household income inequality, then no household can have a negative income. When used as a measure of income inequality, the most unequal society will be one in which a single person receives 100% of the total income and the remaining people receive none (G=1); and the most equal society will be one in which every person receives the same income (G=0).

Some find it more intuitive (and it is mathematically equivalent) to think of the Gini coefficient as half of the relative mean difference. The mean difference is the average absolute difference between two items selected randomly from a population, and the relative mean difference is the mean difference divided by the average, to normalize for scale.

Calculation

The Gini index is defined as a ratio of the areas on the Lorenz curve diagram. If the area between the line of perfect equality and the Lorenz curve is A, and the area under the Lorenz curve is B, then the Gini index is A/(A+B). Since A+B = 0.5, the Gini index, G = A/(0.5) = 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:

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:

G = \frac{1}{n}\left ( n%2B1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n%2B1-i)y_i}{\Sigma_{i=1}^n y_i} \right ) \right )
This may be simplified to:
G = \frac{2 \Sigma_{i=1}^n \; i y_i}{n \Sigma_{i=1}^n y_i} -\frac{n%2B1}{n}
G = 1 - \frac{\Sigma_{i=1}^n \; f(y_i)(S_{i-1}%2BS_i)}{S_n}
where
S_i = \Sigma_{j=1}^i \; f(y_j)\,y_j\, and S_0 = 0\,
G = 1 - \frac{1}{\mu}\int_0^\infty (1-F(y))^2dy = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y))dy
G(S) = \frac{1}{n-1}\left (n%2B1 - 2 \left ( \frac{\Sigma_{i=1}^n \; (n%2B1-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 G, G(S) has a simpler form:
G(S) = 1 - \frac{2}{n-1}\left ( n - \frac{\Sigma_{i=1}^n \; iy_i}{\Sigma_{i=1}^n y_i}\right ) .

There does not exist a sample statistic that is in general an unbiased estimator of the population Gini coefficient, like the relative mean difference.

For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a lognormal distribution with the standard deviation of logs equal to \sigma, then G = 2 \Phi(\sigma/\sqrt{2})-1 where \Phi() is the cumulative distribution function of the standard normal distribution.

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:

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:

G_1 = 1 - \sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} %2B 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.

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 bootstrap techniques but those proposed have been mathematically complicated and computationally onerous even in an era of fast computers. Ogwang (2000) made the process more efficient by setting up a “trick regression model” in which the incomes 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 error term whose variance is inversely proportional to yk;

k = A %2B \ N(0, s^{2}/y_k)

Ogwang showed that G can be expressed as a function of the weighted least squares estimate of the constant A and that this can be used to speed up the calculation of the jackknife estimate for the standard error. Giles (2004) argued that the standard error of the estimate of A can be used to derive that of the estimate of G directly without using a jackknife at all. This method only requires the use of ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improving with increasing sample size. The paper describing this method can be found here: http://web.uvic.ca/econ/ewp0202.pdf

However it has since been argued that this is dependent on the model’s assumptions about the error distributions (Ogwang 2004) and the independence of error terms (Reza & Gastwirth 2006) and that these assumptions are often not valid for real data sets. It may therefore be better to stick with jackknife methods such as those proposed by Yitzhaki (1991) and Karagiannis and Kovacevic (2000). The debate continues.

The Gini coefficient can be calculated if you know the mean of a distribution, the number of people (or percentiles), and the income of each person (or percentile). Princeton development economist Angus Deaton (1997, 139) simplified the Gini calculation to one easy formula:

G = \frac{N%2B1}{N-1}-\frac{2}{N(N-1)u}(\Sigma_{i=1}^n \; P_iX_i)

where u is mean income of the population, Pi 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 Transfer Principle.

Generalised inequality index

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, r_j = x_j / \overline{x}, equals 1 for all j units in some population; for example, there is perfect income equality when everyone’s income x_j equals the mean income \overline{x}, so that r_j=1 for everyone). Measures of inequality, then, are measures of the average deviations of the r_j=1 from 1; the greater the average deviation, the greater the inequality. Based on these observations the inequality indices have this common form:[6]

\mathrm{Inequality} = \Sigma_j \, p_j \, f(r_j)\, ,

where pj weights the units by their population share, and f(rj) is a function of the deviation of each unit’s rj from 1, the point of equality. The insight of this generalised inequality index is that inequality indices differ because they employ different functions of the distance of the inequality ratios (the rj) from 1.

Gini coefficient of income distributions

See also: List of countries by income equality

While developed European nations and Canada tend to have Gini indices between 0.24 and 0.36, the United States' and Mexico's Gini indices are both above 0.40, indicating that the United States (according to the US Census Bureau[7][8][9]) and Mexico have 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 index for the entire world has been estimated by various parties to be between 0.56 and 0.66.[10][11] The graph shows the values expressed as a percentage, in their historical development for a number of countries.

EU Gini index

In 2005 the Gini index for the EU was estimated at 31.[12]

US income Gini indices over time

Gini indices for the United States at various times, according to the US Census Bureau:[13][14][15]

Advantages and disadvantages

Advantages of Gini coefficient as a measure of inequality

The Gini coefficient's main advantage is that it is a measure of inequality by means of a ratio analysis. This makes it easily interpretable, and avoids references to a statistical average or position unrepresentative of most of the population, such as per capita income or gross domestic product. The simplicity of Gini makes it easy to use for comparison across diverse countries and also allows comparison of income distributions across different groups as well as countries; for example the Gini coefficient for urban areas differs from that of rural areas in many countries (though not in the United States). Like any time-based measure, Gini coefficients can be used to compare income distribution over time, thus it is possible to see if inequality is increasing or decreasing independent of absolute incomes. The Gini coefficient satisfies four principles suggested to be important:[16]

Disadvantages of Gini coefficient as a measure of inequality

The limitations of Gini largely lie in its relative nature: It loses information about absolute national and personal incomes. Countries may have identical Gini coefficients, but differ greatly in wealth. Basic necessities may be equal (available to all) in a rich country, while in the poor country, even basic necessities are unequally available.

By measuring inequality in income, the Gini ignores the differential efficiency of 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 (e.g. Sweden) can appear more equal, yet have high Gini coefficients for wealth (for instance 77% of the share value owned by households is held by just 5% of Swedish shareholding households).[17] These factors are not assessed in income-based Gini.

Gini has some mathematical limitations as well. For instance, 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. For a large, economically diverse country, a much higher coefficient will be calculated for the country as a whole than will be calculated for each of its regions. (The coefficient is usually applied to measurable nominal income rather than local purchasing power, tending to increase the calculated coefficient across larger areas.)

As is the case for any single measure of a distribution, economies with similar incomes and Gini coefficients can still have very different income distributions. This results from differing shapes of the Lorenz curve. For example, consider a society where half of individuals had no income and the other half shared all the income equally (i.e. whose Lorenz curve is linear from (0,0) to (0.5,0) and then linear to (1,1)). As is easily calculated, this society has Gini coefficient 0.5 -- the same as that of a society in which 75% of people equally shared 25% of income while the remaining 25% equally shared 75% (i.e. whose Lorenz curve is linear from (0,0) to (0.75,0.25) and then linear to (1,1)).

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. For example, if two equally egalitarian countries pursue different immigration policies, the country accepting a higher proportion of low-income or impoverished migrants will be assessed as less equal (gain a higher Gini coefficient).

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 differential equality when none exist taking into account demographic effects. Thus a given economy may have a higher Gini coefficient at any one point in time compared to another, while the Gini coefficient calculated over individuals' lifetime income is actually lower than the apparently more equal (at a given point in time) economy's.[18] Essentially, what matters is not just inequality in any particular year, but the composition of the distribution over time.

General problems of measurement

As one result of this criticism, in addition to or in competition with the Gini coefficient entropy measures are frequently used (e.g. the Theil Index and the Atkinson index). 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.

Credit risk

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

The discriminatory power refers to a credit risk model's ability to differentiate between defaulting and non-defaulting clients. The above formula G_1 may be used for the final model and also at individual model factor level, to quantify the discriminatory power of individual factors. This is as a result of too many non defaulting clients falling into the lower points scale e.g. factor has a 10 point scale and 30% of non defaulting clients are being assigned the lowest points available e.g. 0 or negative points. This indicates that the factor is behaving in a counter-intuitive manner and would require further investigation at the model development stage.[20]

Relation to other statistical measures

Gini coefficient closely related to the AUC (Area Under receiver operating characteristic Curve) measure of performance[21]. The relation follows the formula AUC = (G%2B1)/2. Gini coefficient is also closely related to Mann–Whitney U.

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 cumulative proportion of individuals.[22] In health, it has been used as a measure of the inequality of health related quality of life in a population.[23] In education, it has been used as a measure of the inequality of universities.[24] In chemistry it has been used to express the selectivity of protein kinase inhibitors against a panel of kinases.[25] In engineering, it has been used to evaluate the fairness achieved by Internet routers in scheduling packet transmissions from different flows of traffic.[26] In statistics, building decision trees, it is used to measure the purity of possible child nodes, with the aim of maximising the average purity of two child nodes when splitting, and it has been compared with other equality measures.[27]

See also

References

  1. ^ Gini, C. (1912) (Italian: Variabilità e mutabilità (Variability and Mutability), C. Cuppini, Bologna, 156 pages. Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).
  2. ^ Gini, C (1909) Concentration and dependency ratios (in Italian). English translation in Rivista di Politica Economica, 87 (1997), 769-789.
  3. ^ "Current Population Survey (CPS) - Definitions and Explanations". US Census Bureau. http://www.census.gov/population/www/cps/cpsdef.html. 
  4. ^ Sadras, V. O.; Bongiovanni, R. (2004). "Use of Lorenz curves and Gini coefficients to assess yield inequality within paddocks". Field Crops Research 90 (2–3): 303–310. doi:10.1016/j.fcr.2004.04.003. 
  5. ^ Gini, C. (1936) On the Measure of Concentration with Special Reference to Income and Statistics, Colorado College Publication, General Series No. 208, 73-79.
  6. ^ Firebaugh, Glenn (1999). "Empirics of World Income Inequality". American Journal of Sociology 104 (6): 1597–1630. doi:10.1086/210218. . See also     (2003). "Inequality: What it is and how it is measured". The New Geography of Global Income Inequality. Cambridge, MA: Harvard University Press. ISBN 0674010671. 
  7. ^ "Gini Ratios for Households, by Race and Hispanic Origin of Householder: 1967 to 2007". Historical Income Tables - Households. United States Census Bureau. http://www.census.gov/hhes/www/income/histinc/h04.html. 
  8. ^ "Table 3. Income Distribution Measures Using Money Income and Equivalence-Adjusted Income: 2007 and 2008". Income, Poverty, and Health Insurance Coverage in the United States: 2008. United States Census Bureau. p. 17. http://www.census.gov/prod/2009pubs/p60-236.pdf. 
  9. ^ "Income, Poverty and Health Insurance Coverage in the United States: 2009". Newsroom. United States Census Bureau. http://www.census.gov/newsroom/releases/archives/income_wealth/cb10-144.html. 
  10. ^ Bob Sutcliffe (April 2007). "Postscript to the article ‘World inequality and globalization’ (Oxford Review of Economic Policy, Spring 2004)". http://siteresources.worldbank.org/INTDECINEQ/Resources/PSBSutcliffe.pdf. Retrieved 2007-12-13 
  11. ^ United Nations Development Programme
  12. ^ "Monitoring quality of life in Europe - Gini index". Eurofound. 26 August 2009. http://www.eurofound.europa.eu/areas/qualityoflife/eurlife/index.php?template=3&radioindic=158&idDomain=3 
  13. ^ "Gini Ratios for Households, by Race and Hispanic Origin of Householder: 1967 to 2007". Historical Income Tables - Households. United States Census Bureau. http://www.census.gov/hhes/www/income/histinc/h04.html. 
  14. ^ "Table 3. Income Distribution Measures Using Money Income and Equivalence-Adjusted Income: 2007 and 2008". Income, Poverty, and Health Insurance Coverage in the United States: 2008. United States Census Bureau. p. 17. http://www.census.gov/prod/2009pubs/p60-236.pdf. 
  15. ^ "Income, Poverty and Health Insurance Coverage in the United States: 2009". Newsroom. United States Census Bureau. http://www.census.gov/newsroom/releases/archives/income_wealth/cb10-144.html. 
  16. ^ Ray, Debraj (1998). Development Economics. Princeton, NJ: Princeton University Press. p. 188. ISBN 0691017069. 
  17. ^ (Data from the Statistics Sweden)
  18. ^ Blomquist, N. (1981). "A comparison of distributions of annual and lifetime income: Sweden around 1970". Review of Income and Wealth 27 (3): 243–264. doi:10.1111/j.1475-4991.1981.tb00227.x. 
  19. ^ Millar, James R. (1987). Politics, work, and daily life in the USSR. New York: Cambridge University Press. p. 193. ISBN 0521348900. 
  20. ^ The Analytics of risk model validation
  21. ^ Hand, David J.; RobersT J. Till (2001). "A Simple Generalisation of the Area Under the ROC Curve for Multiple Class Classification Problems". Machine Learning 45 (2): 171–186. doi:10.1023/A:1010920819831. http://www.springerlink.com/content/nn141j42838n7u21/. 
  22. ^ Wittebolle, Lieven; et al. (2009). "Initial community evenness favours functionality under selective stress". Nature 458 (7238): 623–626. doi:10.1038/nature07840. PMID 19270679. 
  23. ^ Asada, Yukiko (2005). "Assessment of the health of Americans: the average health-related quality of life and its inequality across individuals and groups". Population Health Metrics 3: 7. doi:10.1186/1478-7954-3-7. PMC 1192818. PMID 16014174. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1192818. 
  24. ^ Halffman, Willem; Leydesdorff, L (2010). "Is Inequality Among Universities Increasing? Gini Coefficients and the Elusive Rise of Elite Universities". Minerva 48 (1): 55–72. doi:10.1007/s11024-010-9141-3. PMC 2850525. PMID 20401157. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=2850525. 
  25. ^ Graczyk, Piotr (2007). "Gini Coefficient: A New Way To Express Selectivity of Kinase Inhibitors against a Family of Kinases". Journal of Medicinal Chemistry 50 (23): 5773–5779. doi:10.1021/jm070562u. PMID 17948979. 
  26. ^ Shi, Hongyuan; Sethu, Harish (2003). "Greedy Fair Queueing: A Goal-Oriented Strategy for Fair Real-Time Packet Scheduling". Proceedings of the 24th IEEE Real-Time Systems Symposium. IEEE Computer Society. pp. 345–356. ISBN 0-7695-2044-8. 
  27. ^ Gonzalez, Luis; et al. (2010). "The Similarity between the Square of the Coeficient of Variation and the Gini Index of a General Random Variable". Journal of Quantitative Methods for Economics and Business Administration 10: 5–18. ISSN 1886-516X. http://www.upo.es/RevMetCuant/art.php?id=40. 

Further reading

  • Amiel, Y.; Cowell, F.A. (1999). Thinking about Inequality. Cambridge. ISBN 0521466962. 
  • Anand, Sudhir (1983). Inequality and Poverty in Malaysia. New York: Oxford University Press. ISBN 0195201531. 
  • 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 (9): 1243–1256. doi:10.1016/0277-9536(94)90189-9. PMID 8016689. 
  • Chakravarty, S. R. (1990). Ethical Social Index Numbers. New York: Springer-Verlag. ISBN 0387522743. 
  • Deaton, Angus (1997). Analysis of Household Surveys. Baltimore MD: Johns Hopkins University Press. ISBN 0585237875. 
  • Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. (1987). "Bootstrapping the Gini coefficient of inequality". Ecology (Ecological Society of America) 68 (5): 1548–1551. doi:10.2307/1939238. JSTOR 1939238. 
  • Dorfman, Robert (1979). "A Formula for the Gini Coefficient". The Review of Economics and Statistics (The MIT Press) 61 (1): 146–149. doi:10.2307/1924845. JSTOR 1924845. 
  • Firebaugh, Glenn (2003). The New Geography of Global Income Inequality. Cambridge MA: Harvard University Press. ISBN 0674010671. 
  • Gastwirth, Joseph L. (1972). "The Estimation of the Lorenz Curve and Gini Index". The Review of Economics and Statistics (The MIT Press) 54 (3): 306–316. doi:10.2307/1937992. JSTOR 1937992. 
  • Giles, David (2004). "Calculating a Standard Error for the Gini Coefficient: Some Further Results". Oxford Bulletin of Economics and Statistics 66 (3): 425–433. doi:10.1111/j.1468-0084.2004.00086.x. 
  • 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 of Incomes". The Economic Journal (Blackwell Publishing) 31 (121): 124–126. doi:10.2307/2223319. JSTOR 2223319. 
  • Giorgi, G. M. (1990). A bibliographic portrait of the Gini ratio, Metron, 48, 183-231.
  • Karagiannis, E. and Kovacevic, M. (2000). "A Method to Calculate the Jackknife Variance Estimator for the Gini Coefficient". Oxford Bulletin of Economics and Statistics 62: 119–122. doi:10.1111/1468-0084.00163. 
  • Mills, Jeffrey A.; Zandvakili, Sourushe (1997). "Statistical Inference via Bootstrapping for Measures of Inequality". Journal of Applied Econometrics 12 (2): 133–150. doi:10.1002/(SICI)1099-1255(199703)12:2<133::AID-JAE433>3.0.CO;2-H. 
  • Modarres, Reza and Gastwirth, Joseph L. (2006). "A Cautionary Note on Estimating the Standard Error of the Gini Index of Inequality". Oxford Bulletin of Economics and Statistics 68 (3): 385–390. doi:10.1111/j.1468-0084.2006.00167.x. 
  • Morgan, James (1962). "The Anatomy of Income Distribution". The Review of Economics and Statistics (The MIT Press) 44 (3): 270–283. doi:10.2307/1926398. JSTOR 1926398. 
  • Ogwang, Tomson (2000). "A Convenient Method of Computing the Gini Index and its Standard Error". Oxford Bulletin of Economics and Statistics 62: 123–129. doi:10.1111/1468-0084.00164. 
  • Ogwang, Tomson (2004). "Calculating a Standard Error for the Gini Coefficient: Some Further Results: Reply". Oxford Bulletin of Economics and Statistics 66 (3): 435–437. doi:10.1111/j.1468-0084.2004.00087.x. 
  • Xu, Kuan (January 2004). How Has the Literature on Gini's Index Evolved in the Past 80 Years?. Department of Economics, Dalhousie University. http://economics.dal.ca/RePEc/dal/wparch/howgini.pdf. Retrieved 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. 
  • Yitzhaki, S. (1991). "Calculating Jackknife Variance Estimators for Parameters of the Gini Method". Journal of Business and Economic Statistics (American Statistical Association) 9 (2): 235–239. doi:10.2307/1391792. JSTOR 1391792. 

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