Gini Coefficient for Grouped Data

Computes the Gini coefficient from grouped income data based on linear interpolation of income shares.

Usage

gini_grouped(Y, freq)

Arguments

Y: Numeric vector of total amounts per group. Each element represents the sum of the variable of interest (e.g. income, wealth, consumption) for all individuals or units in that group.
freq: Numeric vector of frequencies (number of individuals or units) per group. Must have the same length as Y.

Value

A numeric value representing the estimated Gini coefficient on grouped data. The Gini coefficient ranges from 0 (perfect equality) to 1 (complete inequality). Note that it assumes equality within groups.

Details

Consider grouped data divided into $L$ classes with known boundaries, observed frequencies $f_1, \ldots, f_L$ and total amounts $Y_1, \ldots, Y_J$. The Gini coefficient is approximated by linear interpolation of cumulative shares, as:

$$G \approx 1 - \sum_{j=1}^{J} (s_j + s_{j-1})(u_j - u_{j-1})$$

where:

$p_j = f_j / \sum_{i=1}^{J} f_i$ is the population share of group $j$;
$c_j = Y_j / \sum_{i=1}^{J} Y_i$ is the share of the variable of interest in group $j$;
$s_j = \sum_{k=1}^{j} c_k$ is the cumulative share of the variable up to group $j$;
$u_j = \sum_{k=1}^{j} p_k$ is the cumulative population share up to group $j$;
$s_0 = u_0 = 0$ by convention.

This formula computes twice the area between the egalitarian line (perfect equality) and the Lorenz curve obtained by linearly interpolating the points $(u_j, s_j)$. Since it assumes all observations within a group have identical values, it provides a lower-bound estimate of the true Gini coefficient – actual inequality may be larger (Jorda et al. 2021) .

Note

Important limitations:

This is a lower bound approximation. The true Gini coefficient may be higher due to within-group inequality.
The bias magnitude depends on the number of groups and how they are defined.
Groups should ideally be defined to maximize within-group homogeneity.

References

Jorda V, Sarabia JM, Jäntti M (2021). “Inequality measurement with grouped data: Parametric and non-parametric methods.” Journal of the Royal Statistical Society Series A: Statistics in Society, 184(3), 964–984.

Examples

income_freq <- c(1200, 1800, 1500, 800, 400, 20, 10)
income_tot <- c(18800, 16300, 44700, 33900, 21500, 22100, 98300)

gini_grouped(Y = income_tot, freq = income_freq)
#> [1] 0.6201874