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Graphical representation of the Gini coefficient(The area of the whole triangle is defined as 1.)
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. In Statistics, (statistical dispersion (also called statistical variability or variation) is variability or spread in a Variable or a Probability Income inequality metrics or income distribution metrics are techniques used by economists to measure the distribution of Income and Economic inequality 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 It is defined as a ratio with values between 0 and 1: A low Gini coefficient indicates more equal income or wealth distribution, while a high Gini coefficient indicates more unequal distribution. A ratio is an expression which compares quantities relative to each other 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. Worldwide, Gini coefficients range from approximately 0. 249 in Japan to 0. 707 in Namibia.

As opposed to the Gini coefficient, the Gini index is the Gini coefficient expressed as a percentage, and is equal to the Gini coefficient multiplied by 100. The Gini index is more widely used, for example in country listings in Wikipedia.

The Gini coefficient was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variability and Mutability" (Italian: Variabilità e mutabilità ). The' Italian people' are a Southern European Ethnic group located primarily in Italy, Switzerland, France and by virtue of a wide-ranging Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. Corrado Gini ( May 23, 1884 - March 13, 1965) was an Italian Statistician, demographer and sociologist Year 1912 ( MCMXII) was a Leap year starting on Monday (link will display the full calendar of the Gregorian calendar (or a Leap year starting Italian ( or lingua italiana) is a Romance language spoken by about 63 million people as a First language, primarily in Italy.

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 debtor's non-payment of a Loan or other line of credit (either the principal or Interest (coupon or both Faced Since gini coefficient addresses wealth inequality it may be important to understand what a transformative asset is. Transformative assets are assets that may provide resources for social and Economic mobility. Transformative assets increase the gini coefficient as they provide a family or individual with a wealth advantage over most persons.


Contents

Calculation

The Gini coefficient is defined as a ratio of the areas on the Lorenz curve diagram. The Lorenz curve is a graphical representation of the Cumulative distribution function of a Probability distribution; it is a graph showing the proportion 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/(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. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space For example:

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 )
This may be simplified to:
G = \frac{2 \Sigma_{i=1}^n \; i y_i}{n \Sigma_{i=1}^n y_i} -\frac{n+1}{n}
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 = \frac{1}{\mu}\int_0^\infty F(y)(1-F(y))dy
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. In Statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population Parameter (which is called the In Statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population Parameter (which is called the In Statistics, an estimator is a function of the observable sample data that is used to estimate an unknown population Parameter (which is called the 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. The mean difference is a measure of statistical dispersion 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. In the mathematical subfield of Numerical analysis, interpolation is a method of constructing new data points within the range of a Discrete set of 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} + Y_{k-1})

is the resulting approximation for G. In Mathematics, the trapezium rule (the British term or trapezoidal rule (the American term is a way to approximately calculate the definite integral 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. In Numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite Integral, and by extension In Numerical analysis, Simpson's rule is a method for Numerical integration, the numerical approximation of Definite integrals Specifically it is the following 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 + \ 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 calculaton of the jackknife esimate for the standard error. The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields 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. 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.

Income Gini indices 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. This is a list of countries or dependencies by Income inequality metrics, including Gini coefficients according to the United Nations (UN and the Economic inequality refers to disparities in the distribution of Economic Assets and Income.

Gini coefficient, income distribution by country.
Gini coefficient, income distribution by country.

While most developed European nations tend to have Gini indices between 24 and 36, the United States' and Mexico's Gini indices are both above 40, indicating that the United States and Mexico have greater inequality. The United States of America —commonly referred to as the The economy of Mexico is 10th to 12th largest in the worldSince the 1994 crisis, administrations have improved the country's macroeconomic fundamentals. Using the Gini can help quantify differences in welfare and compensation policies and philosophies. "Social welfare" redirects here For other uses see Welfare A social welfare provision refers to any program which seeks to provide Living wage is a term used to describe the minimum hourly wage necessary for a person to achieve some specific standard of living 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 is a measure of statistical dispersion most prominently used as a measure of inequality of income distribution or inequality of wealth

The Gini index for the entire world has been estimated by various parties to be between 56 and 66. [1][2]


Gini indices, income distribution over time for selected countries
Gini indices, income distribution over time for selected countries

Correlation with per-capita GDP

Poor countries (those with low per-capita GDP) have Gini indices that fall over the whole range from low (25) to high (71), while rich countries generally have intermediate Gini indices (under 40). This article includes three lists of Countries of the world sorted by their Gross domestic product (GDP at Purchasing power parity (PPP Per capita The lowest Gini coefficients can be found in Japan, Scandinavian countries, and in many recently ex-socialist countries of Eastern Europe. For a topic outline on this subject see List of basic Japan topics. Terminology and usage As a cultural term "Scandinavia" has no official definition and is subject to usage by those who identify with the culture in question as well Eastern Europe is a general term that refers to the Geopolitical region encompassing the easternmost part of the European continent. Note that in many of the former socialist countries, the sizeable underground economy hides income for many. In such a case, earning/wealth statistics over-represent certain income ranges (i. e. , in lower-income regions), and may decrease the Gini coefficient even in the presence of real inequality.

US income Gini indices over time

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

Advantages of Gini coefficient as a measure of inequality

Disadvantages of Gini coefficient as a measure of inequality

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. The European Union ( EU) is a political and economic union of twenty-seven member states, located primarily in 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.

Problems in using the Gini coefficient

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 index of Atkinson). The Theil index, derived by econometrician Henri Theil, is a statistic used to measure Economic inequality. Sir Anthony Barnes "Tony" Atkinson, FBA is a British Economist and has been a Senior Research Fellow of Nuffield College Oxford since 2005 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. Probability is the likelihood or chance that something is the case or will happen

See also

Notes

  1. ^ Bob Sutcliffe (2007), Postscript to the article ‘World inequality and globalization’ (Oxford Review of Economic Policy, Spring 2004), <http://siteresources.worldbank.org/INTDECINEQ/Resources/PSBSutcliffe.pdf>. This is a list of countries by Human Development Index as included in the United Nations Development Program 's Human Development Report 2007 The Human Poverty Index is an indication of the Standard of living in a country developed by the United Nations (UN The Pareto distribution, named after the Italian Economist Vilfredo Pareto, is a Power law Probability distribution that coincides with In Signal detection theory, a receiver operating characteristic ( ROC) or simply ROC curve, is a graphical plot of the sensitivity "Social welfare" redirects here For other uses see Welfare A social welfare provision refers to any program which seeks to provide Welfare economics is a branch of Economics that uses microeconomic techniques to simultaneously determine Allocative efficiency within an economy and the The Atkinson index (also known as the Atkinson measure) is a measure of Economic Income inequality developed by Anthony Barnes Atkinson. The Theil index, derived by econometrician Henri Theil, is a statistic used to measure Economic inequality. The Robin Hood index, also known as the Hoover index, is a measure of income inequality. The Suits index of a public policy is a measure of collective progressivity named for Economist Daniel B Income inequality metrics or income distribution metrics are techniques used by economists to measure the distribution of Income and Economic inequality 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 Retrieved on 2007-12-13 
  2. ^ United Nations Development Programme
  3. ^ Note that the calculation of the index for the United States was changed in 1992, resulting in an upwards shift of about 2. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1294 - Saint Celestine V abdicates the papacy after only five months Celestine hoped to return to his previous life
  4. ^ European Union, CIA World Factbook, <https://www.cia.gov/library/publications/the-world-factbook/geos/ee.html>. Retrieved on 2227-12-13 
  5. ^ United States, CIA World Factbook, <https://www.cia.gov/library/publications/the-world-factbook/geos/us.html>. The 23rd century of the Anno Domini ( common) era will span the years 2201&ndash2300 of the Gregorian calendar. Events 1294 - Saint Celestine V abdicates the papacy after only five months Celestine hoped to return to his previous life Retrieved on 2227-12-13 
  6. ^ Friedman, David D.
  7. ^ (Data from the Statistics Sweden. The 23rd century of the Anno Domini ( common) era will span the years 2201&ndash2300 of the Gregorian calendar. Events 1294 - Saint Celestine V abdicates the papacy after only five months Celestine hoped to return to his previous life )

References

External links

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