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 and the "most unequal" society (composed of N individuals) where a single person receives 100% of the total income and the remaining  people receive none.

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 yi, i = 1 to n, indexed in non-decreasing order (yi ≤ yi+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:
 * $$G = \frac{2 \sum_{i=1}^n i y_i}{n \sum_{i=1}^n y_i} -\frac{n+1}{n}.$$
 * This formula actually applies to any real population, since each person can be assigned his or her own yi.

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 yi, i = 1 to n, that are indexed in non-decreasing order (yi ≤ yi+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, 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.