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Encyclopedia > Gini coefficient
Graphical representation of the Gini coefficient

The Gini coefficient is 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 (e.g. everyone has the same income) and 1 corresponds to perfect inequality (e.g. 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. Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... Income inequality metrics or income distribution metrics are techniques used by economists to measure the distribution of income among the participants in a particular economy, such as that of a specific country or of the world in general. ... Wealth condensation is a theoretical process by which, in certain conditions, newly-created wealth tends to become concentrated in the possession of already-wealthy individuals or entities. ... A ratio is a quantity that denotes the proportional amount or magnitude of one quantity relative to another. ... The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. ...

The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability"). A graph of a normal bell curve showing statistics used in educational assessment and comparing various grading methods. ... Corrado Gini (May 23, 1884 - March 13, 1965) was an Italian statistician, demographer and sociologist who developed the Gini coefficient, a measure of the income inequality in a society. ... 1912 (MCMXII) was a leap year starting on Monday in the Gregorian calendar (or a leap year starting on Tuesday in the 13-day-slower Julian calendar). ...

The Gini coefficient is also commonly used for the measurement of the discriminatory power of rating systems in credit risk management. A credit rating assesses the credit worthiness of an individual, corporation, or even a country. ... Credit risk is the risk of loss due to a debtors non-payment of a loan or other line of credit (either the principal or interest (coupon) or both). ...

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 percent sign. ... The mean difference is a statistical measure of dispersion and is equal to the average absolute difference of two independent values drawn from a probability distribution. ...

## 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: The Lorenz curve is a graphical representation of the cumulative distribution function of a probability distribution; it is a graph showing the proportion of the distribution assumed by the bottom y% of the values. ... The integral of f(x) from a to b is the area above the x-axis and below the curve y = f(x), minus the area below the x-axis and above the curve, for x in the interval [a,b]. Integration is a core concept of advanced mathematics, specifically...

$G = 1 - 2,int_0^1 L(X) dX$

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):
$G = frac{1}{n}left ( n+1 - 2 left ( frac{Sigma_{i=1}^n ; (n+1-i)y_i}{Sigma_{i=1}^n y_i} right ) right )$
• 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):
$G = 1 - frac{Sigma_{i=1}^n ; f(y_i)(S_{i-1}+S_i)}{S_n}$
where:
$S_i = Sigma_{j=1}^i ; f(y_j),y_j,$ and $S_0 = 0,$
$G = 1 - frac{1}{mu}int_0^infty (1-F(y))^2dy$

Since the Gini coefficient is half the relative mean difference, it can also be calculated using formulas for the relative mean difference. In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... The mean difference is a statistical measure of dispersion and is equal to the average absolute difference of two independent values drawn from a probability distribution. ...

For a random sample S consisting of values yi, i = 1 to n, that are indexed in non-decreasing order ( yiyi+1), the statistic:

$G(S) = frac{1}{n-1}left (n+1 - 2 left ( frac{Sigma_{i=1}^n ; (n+1-i)y_i}{Sigma_{i=1}^n y_i}right ) right )$

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. In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... In statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population parameter; an estimate is the result from the actual application of the function to a particular set of data. ... The mean difference is a statistical measure of dispersion and is equal to the average absolute difference of two independent values drawn from a probability 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: In the mathematical subfield of numerical analysis, interpolation is a method of constructing new data points from a discrete set of known data points. ...

• 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: The function f(x) (in blue) is approximated by a linear function (in red). ...

$G_1 = 1 - sum_{k=1}^{n} (X_{k} - X_{k-1}) (Y_{k} + Y_{k-1})$

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. Numerical Integration with the Monte Carlo method: Nodes are random equally distributed. ... The function f(x) (in blue) is approximated by a quadratic function P(x) (in red). ...

## 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. World map of the Gini coefficient This is a list of countries or dependencies by Income inequality metrics, sorted in ascending order according to their Gini coefficient. ... Differences in national income equality around the world as measured by the national Gini coefficient. ...

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). Image File history File links This is a lossless scalable vector image. ... Image File history File links This is a lossless scalable vector image. ... ... Living wage refers to the minimum hourly wage necessary for a person to achieve a basic standard of living. ... Graphical representation of the Gini coefficient The Gini coefficient is a measure of inequality of income distribution or inequality of wealth distribution. ...

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

Gini coefficients, income distribution over time for selected countries

Image File history File links Download high resolution version (950x688, 60 KB) Summary Gini coefficients of selected countries, from publicly available data from the World Bank, Nationmaster, and the US Census Bureau. ... Image File history File links Download high resolution version (950x688, 60 KB) Summary Gini coefficients of selected countries, from publicly available data from the World Bank, Nationmaster, and the US Census Bureau. ...

### 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. Map of countries by GDP (PPP) per capita for the year 2006. ...

### US income gini coefficients over time

Gini coefficients for the United States at various times, according to the US Census Bureau: The United States Census Bureau (officially Bureau of the Census as defined in Title ) is a part of the United States Department of Commerce. ...

• 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. Year 1967 (MCMLXVII) was a common year starting on Sunday and the summer of 1967 was known as The Summer of Peace and Love (link will display full calendar) of the 1967 Gregorian calendar. ... Year 1968 (MCMLXVIII) was a leap year starting on Monday (link will display full calendar) of the Gregorian calendar. ... 1970 (MCMLXX) was a common year starting on Thursday. ... Year 1980 (MCMLXXX) was a leap year starting on Tuesday (link displays the 1980 Gregorian calendar). ... Year 1990 (MCMXC) was a common year starting on Monday (link displays the 1990 Gregorian calendar). ... 2000 (MM) was a leap year starting on Saturday of the Gregorian calendar. ... Year 2005 (MMV) was a common year starting on Saturday (link displays full calendar) of the Gregorian calendar. ...

## 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.

In number and more generally in algebra, a ratio is the linear relationship between two quantities. ... The per capita income for a group of people may be defined as their total personal income, divided by the total population. ... Nominal GDP per person (capita) in 2006. ...

## 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 T converted by 1 − e T into a inequality coefficient) averts such problems. The Theil index, derived by econometrician Henri Theil, is a statistic used to measure economic inequality. ...

• 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 don't 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.

Egalitarianism (derived from the French word Ã©gal, meaning equal or level) is a political doctrine that holds that all people should be treated as equals from birth. ...

## 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. The Food Stamp Program serves as the first line of defense against hunger. ... In economics, the Atkinson index or Atkinson measure is used to quantify income inequality. ... The Theil index, derived by econometrician Henri Theil, is a statistic used to measure economic inequality. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ... The word probability derives from the Latin probare (to prove, or to test). ...

## 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

• 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.

Year 2006 (MMVI) was a common year starting on Sunday (link displays full 2006 calendar) of the Gregorian calendar. ... June 1 is the 152nd day of the year (153rd in leap years) in the Gregorian calendar. ...

World map of the Gini coefficient This is a list of countries or dependencies by Income inequality metrics, sorted in ascending order according to their Gini coefficient. ... Coloured world map indicating Human Development Index (2004) (colour-blind compliant map) This is a list of countries by Human Development Index as included in the United Nations Development Programmes Human Development Report 2006, compiled on the basis of 2004 data. ... The Human Poverty Index is an indication of the standard of living in a country, developed by the United Nations (UN). ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution associated with it. ... Income inequality metrics or income distribution metrics are techniques used by economists to measure the distribution of income among the participants in a particular economy, such as that of a specific country or of the world in general. ... Receiver operating characteristic analysis (ROC analysis) provides tools to select possibly optimal models and to discard suboptimal ones independently from (and prior to specifying) the cost context or the class distribution. ... Social welfare can be taken to mean the welfare or well-being of a society. ... The Pareto distribution, named after the Italian economist Vilfredo Pareto, is a power law probability distribution found in a large number of real-world situations. ... The Robin Hood index is a measure of income inequality. ... The mean difference is a statistical measure of dispersion and is equal to the average absolute difference of two independent values drawn from a probability distribution. ... The Suits index of a public policy is a measure of collective progressivity. ...

Results from FactBites:

 CHAPTER 9: REDUCTIONS IN INEQUALITY (1420 words) The Gini coefficient measures the area between the straight line connecting (0,0) and (1,1) and the Lorenz curve connecting those two points, as a proportion of the triangle formed by (0,0), (1,0), and (1,1). The Gini coefficient may also be used to calibrate the effects of work and school on inequality. The Gini coefficient for household broadband declined from.395 in August 2000 to.374 in September 2001.
 Gini coefficient - Wikipedia, the free encyclopedia (1300 words) The Gini coefficient is a measure of inequality of a distribution, defined as the ratio of area between the Lorenz curve of the distribution and the curve of the uniform distribution, to the area under the uniform distribution. 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 defined as a ratio of the areas on the Lorenz curve diagram.
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