Editing Gini coefficient (section)

=== 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 of ''n'' individuals with values <math>y_1 \leq y_2\leq \cdots \leq y_n </math>,<ref name="Wolfram Mathworld">{{cite web |title=Gini Coefficient |url=http://mathworld.wolfram.com/GiniCoefficient.html |publisher=Wolfram Mathworld}}</ref> 
::<math>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 ). </math>
:This may be simplified to:
::<math>G = \frac{2 \sum_{i=1}^n i y_i}{n \sum_{i=1}^n y_i} -\frac{n+1}{n}.</math>

The Gini coefficient can also be considered as half the [[relative mean absolute difference]]. For a random sample ''S'' with values <math>y_1 \leq y_2\leq \cdots \leq y_n </math>, the sample Gini coefficient
:<math>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 )</math>

is a [[consistent estimator]] of the population Gini coefficient, but is not in general [[estimator#Point estimators|unbiased]]. In simplified form:

:<math>G(S) = 1 - \frac{2}{n-1}\left ( n - \frac{\sum_{i=1}^n iy_i}{\sum_{i=1}^n y_i}\right ). </math>

There does not exist a sample statistic that is always an unbiased estimator of the population Gini coefficient.