User:Jmjosh90/Gini coefficient

While the income distribution of any particular country will not correspond perfectly to the theoretical models, these models can provide a qualitative explanation of the income distribution in a nation given the Gini coefficient.

Example: two levels of income

The extreme cases are represented by the "most equal" society in which every person receives the same income (G = 0) and the "most unequal" society (composed of N individuals) where a single person receives 100% of the total income and the remaining N − 1 people receive none (G = 1 − 1/N).

A simplified case distinguishes just two levels of income, low and high. If the high income group is a proportion u of the population and earns a proportion f of all income, then the Gini coefficient is f − u. A more graded distribution with these same values u and f will always have a higher Gini coefficient than f − u.

An example case in which the wealthiest 20% of the population has 80% of all income (see Pareto principle) would lead to an income Gini coefficient of at least 60%.

Another example case, in which 1% of the world's population owns 50% of all wealth, would result in a wealth Gini coefficient of at least 49%.

Alternative expressions

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

• For a population uniform on the values y_i, i = 1 to n, indexed in non-decreasing order (y_i ≤ y_i+1):

$G={\frac {1}{n}}\left(n+1-2\left({\frac {\sum _{i=1}^{n}(n+1-i)y_{i}}{\sum _{i=1}^{n}y_{i}}}\right)\right).$

This may be simplified to:

$G={\frac {2\sum _{i=1}^{n}iy_{i}}{n\sum _{i=1}^{n}y_{i}}}-{\frac {n+1}{n}}.$

This formula actually applies to any real population, since each person can be assigned his or her own y_i.

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 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:

$G(S)={\frac {1}{n-1}}\left(n+1-2\left({\frac {\sum _{i=1}^{n}(n+1-i)y_{i}}{\sum _{i=1}^{n}y_{i}}}\right)\right)$

is a consistent estimator of the population Gini coefficient, but is not, in general, unbiased. Like G, G(S) has a simpler form:

$G(S)=1-{\frac {2}{n-1}}\left(n-{\frac {\sum _{i=1}^{n}iy_{i}}{\sum _{i=1}^{n}y_{i}}}\right).$

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