Gini coefficient

In , also a Gini index as well as the Gini ratio, is the measure of statistical dispersion included to symbolize the income inequality or the wealth inequality within a nation or a social group. The Gini coefficient was developed by the statistician as alive as sociologist Corrado Gini.

The Gini coefficient measures the inequality among values of a frequency distribution, like the levels of income. A Gini coefficient of 0 expresses perfect equality, where all values are the same i.e. where everyone has the same income while a Gini coefficient of 1 or 100% expresses maximal inequality among values i.e. for a large number of people where only one grown-up has any the income or consumption and all others produce none, the Gini coefficient will be most one.

The Gini coefficient was exposed by Corrado Gini as a degree of inequality of income or wealth. For OECD countries, in the behind 20th century, considering the effect of taxes and transfer payments, the income Gini coefficient ranged between 0.24 and 0.49, with Slovenia being the lowest and Mexico the highest. African countries had the highest pre-tax Gini coefficients in 2008–2009, with South Africa the world's highest, variously estimated to be 0.63 to 0.7, although this figure drops to 0.52 after social support is taken into account, and drops again to 0.47 after taxation. The global income Gini coefficient in 2005 has been estimated to be between 0.61 and 0.68 by various sources.

There are some issues in interpreting a Gini coefficient. The same advantage may a thing that is said from many different distribution curves. The demographic sorting should be taken into account. Countries with an aging population, or with a baby boom, experience an increasing pre-tax Gini coefficient even if real income distribution for working adults manages constant. Scholars create devised over a dozen variants of the Gini coefficient.

Calculation

While the income distribution of any particular country will not always adopt theoretical models in reality, these functions provide a qualitative understanding of the income distribution in a nation assumption the Gini coefficient.

The extreme cases are represented by the "most equal" society in which every grownup receives the same income = 0 and the "most unequal" society composed of N individuals where a single person receives 100% of the or situation. income and the remaining − 1 people get none .

A more general simplified issue also just distinguishes two levels of income, low and high. if the high income chain is a proportion u of the population and earns a proportion f of all income, then the Gini coefficient is . An actual more graded distribution with these same values u and f will always have a higher Gini coefficient than .

The proverbial case where the richest 20% have 80% of all income see Pareto principle would lead to an income Gini coefficient of at least 60%.

The often cited case in which 1% of all the world's population owns 50% of all wealth, would mean a wealth Gini coefficient of at least 49%.

In some cases, this equation can be applied to calculate the Gini coefficient without direct point of reference to the Lorenz curve. For example, taking y to indicate the income or wealth of a person or household:

Since the Gini coefficient is half the relative mean absolute difference, it can also be calculated using formulas for the relative mean absolute difference. For a random pattern S consisting of values y_i, i = 1 to n, that are indexed in non-decreasing structure y_i ≤ y_i+1, the statistic:

is a unbiased. Like G, has a simpler form:

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

For a discrete probability distribution with probability mass function , where is the fraction of the population with income or wealth , the Gini coefficient is:

where

where

When the population is large, the income distribution may be represented by a non-stop probability density function fx where fx dx is the fraction of the population with wealth or income in the interval dx about x. If Fx is the cumulative distribution function for fx:

and Lx is the Lorenz function:

then the Lorenz curve LF may then be represented as a function parametric in Lx and Fx and the expediency of B can be found by integration:

The Gini coefficient can also be calculated directly from the cumulative distribution function of the distribution Fy. instituting μ as the mean of the distribution, and specifying that Fy is zero for all negative values, the Gini coefficient is condition by:

The latter result comes from integration by parts. Note that this formula can be applied when there are negative values if the integration is taken from minus infinity to plus infinity.

The Gini coefficient may be expressed in terms of the quantile function QF inverse of the cumulative distribution function: QFx = x

Since the Gini coefficient is exponential distribution, which is a function of only x and a scale parameter, the Gini coefficient is a constant, equal to 1/2.

For some functional forms, the Gini index can be calculated explicitly. For example, if y follows a error function since , where is the cumulative distribution function of a specifications normal distribution. In the table below, some examples for probability density functions with help on are shown.[] The Dirac delta distribution represents the case where programs has the same wealth or income; it implies that there are no variations at all between incomes.

Sometimes the entire Lorenz curve is not known, and only values atintervals 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, Y_k are the call 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 regarded and noted separately. interval as a line between consecutive points, then the area B can be approximated with trapezoids and:

is the resulting approximation for G. More accurate results can be obtained using other methods to quadratic function across pairs of intervals, or building an appropriately smooth approximation to the underlying distribution function that matches the asked data. If the population mean and boundary values for used to refer to every one of two or more people or things 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 standards error, or confidence intervals for the population Gini coefficient, should be reported. These can be calculated using Tomson Ogwang presents the process more able by determining up a "trick regression model" in which respective income variables in the sample are ranked with the lowest income being planned 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 y_k:

Thus, G can be expressed as a function of the weighted jackknife estimate for the standard error. Economist David Giles 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 ownership of ordinary least squares regression after ordering the sample data. The results compare favorably with the estimates from the jackknife with agreement improved with increasing sample size.

However, it has since been argued that this is dependent on the model's assumptions about the error distributions and the independence of error terms, assumptions that are often not valid for real data sets. There is still ongoing debate surrounding this topic.

Guillermina Jasso and Angus Deaton independently proposed the coming after or as a result of. formula for the Gini coefficient:

where is mean income of the population, P_i 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 helps higher weight to poorer people in the income distribution, which allowed the Gini to meet the Transfer Principle. Note that the Jasso-Deaton formula rescales the coefficient so that its value is 1 if all the are zero apart from one. Note however Allison'son the need to divide by N² instead.

FAO explains another explanation of the formula.