Gini coefficient

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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. It is often used to measure income inequality. It is a number between 0 and 1, where 0 corresponds to perfect equality (i.e. everyone has the same income) and 1 corresponds to perfect inequality (i.e. one person has all the income, while everyone else has zero income). It was developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità" ("Variability and Mutability"). The Gini coefficient is equal to half of the relative mean difference. The Gini index is the Gini coefficient expressed as a percentage, and is equal to the Gini coefficient multiplied by 100.

While the Gini coefficient is mostly used to measure income inequality, it can also be used to measure wealth inequality. This use requires that no one has a negative net wealth.

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:

[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 with values y_i, i = 1 to n, that are indexed in non-decreasing order ( y_i ≤ y_i+1):

[G = \frac(n+1 - 2\,\frac^n \; (n+1-i)y_i}^n y_i})\,]

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

[G = 1 - \frac^n \; f(y_i)(S_+S_i)}]

where:

[S_i = \Sigma_^i \; f(y_j)\,y_j\,] and [S_0 = 0\,]

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

[G = 1 - \frac\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.

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

[G(S) = \frac(n+1 - 2\,\frac^n \; (n+1-i)y_i}^n y_i})\,]

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 technigues for interpolating the missing values of the Lorenz curve. If ( X _k , Y_k ) are the known points on the Lorenz curve, with the X _k indexed in increasing order ( X _{k - 1} < X _k ), so that:

• X_k is the cumulated proportion of the population variable, for k = 0,...,n, with X₀ = 0, X_n = 1.
• Y_k is the cumulated proportion of the income variable, for k = 0,...,n, with Y₀ = 0, Y_n = 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:

[G_1 = 1 - \sum_^ (X_ - X_) (Y_ + Y_)]

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

See complete listing in list of countries by income equality.

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).

US income gini coefficients over time

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

• 1970: 0.394
• 1980: 0.403
• 1990: 0.428
• 2000: 0.462 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].)

Advantages 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, 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:
• * 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 as a measure of inequality

• The Gini coefficient measured for a large geographically 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 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.

• It is claimed that the Gini coefficient is more sensitive to the income of the middle classes than to that of the extremes.

• 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.

Notes

References

• Dixon, PM, Weiner J., Mitchell-Olds T, Woodley R. Bootstrapping the Gini coefficient of inequality. Ecology 1987;68:1548-1551.
• Gini C. "Variabilità e mutabilità" (1912) Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955).

See also

• List of countries by income equality
• List of countries by Human Development Index
• Human Poverty Index
• Welfare economics
• Income inequality metrics
• ROC analysis
• Social welfare (political science)
• Pareto distribution
• Robin Hood index
• Relative mean difference

External links

• UN Human Development [Report 2004, p50-53]: Gini Index calculated for all countries.
• [link] Comparison of Urban and Rural Areas' Gini Coefficients Within States
• [link] World Income Inequality Database
• [link], Forbes Article, In praise of inequality
• [Gini index map from WorldPolicy.org]

• Software:
• * [Free Online Calculator] computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
• * Free Calculator: [Online] and [downloadable scripts] (Python and Lua) for Atkinson, Gini, Hoover and Kullback-Leibler inequalities
• * Users of the [R] data analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.

 

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