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The Gini coefficient is one of the most widely used statistical measures of inequality within a society. It is based on the Lorenz curve, which graphically represents the cumulative distribution of income or consumption across a population. The functional form of the Gini coefficient is expressed as the ratio of the area between the line of perfect equality and the Lorenz curve, to the total area under the line of perfect equality. Mathematically, it can be written as: $$ G = \frac{\sum_{i=1}^n \sum_{j=1}^n |y_i - y_j|}{2n^2 \mu} $$ where $y_i$ and $y_j$ represent the income of individuals $i$ and $j$, $n$ is the total number of individuals in the population, and $\mu$ is the mean income. An alternative continuous formulation is: $$ G = 1 - 2\int_0^1 L(p) dp $$ where $L(p)$ is the Lorenz curve showing the cumulative share of income earned by the bottom ________ _________ _______ ______ ________ ______ _______ ______.
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