#### 3.1.4 Measures of Inequality

##### Lorenz curve

Lorenz curve is an important tool for the analysis of inequality in a variety of situations – inequality in the distribution of income within a population, inequality in the productivity of scientists in a give population, inequality in the allocation of research grants to different institutions by a funding agency, etc.

Lorenz curve is defined as follows.

Consider a sample of n individuals. Let xi denote the income of individual i (i= 1, 2, ... , n), such that x1 £ x2 £ x3 £ . . . £ xn. The sample Lorenz curve is the polygon joining the points (h/n, Lh/Ln), where L0 = 0 and is the total income of the poorest h individuals.

The Lorenz curve q = L(y) has as its abscissa the cumulative proportion of income receivers, ordered by increasing size of their income received. The Lorenz curve can be mathematically represented as:

where Y is a nonnegative income variable for which the mathematical expectation m = E (Y) exists and p = F(y) is the cumulative distribution function of the population of income receivers. When all the members of the population receive the same income, the Lorenz curve is the equidistribution or identity function F = L. The Lorenz curve bends downward to the right, as the distribution of income becomes more unequal.

##### Gini coefficient

Gini coefficient is related to the Lorenz Curve. Gini coefficient is twice the area between the Lorenz curve and the equidistribution function. Gini coefficient ranges between 0 and 1.

0 Þ perfect equality

1 Þ perfect inequality