Theilindex

The Theil index is a statistical measure of inequality commonly used in economics to quantify how uneven a distribution of income or wealth is. Named after Henning Theil, it belongs to the family of generalized entropy measures. Two widely used variants are Theil T and Theil L, which differ in sensitivity to different parts of the distribution and in their decomposability by subgroups.

Let y_i denote the income of individual i, with Y = sum_i y_i and n individuals. The mean income is y_bar = Y / n. The Theil index T is defined as T = (1/n) sum_i ( (y_i / y_bar) ln( y_i / y_bar ) ), using the natural logarithm. Theil L is defined as L = (1/n) sum_i ln( y_bar / y_i ), which can also be written as L = -(1/n) sum_i ln( y_i / y_bar ). Both T and L are nonnegative, with T = 0 and L = 0 if and only if all incomes are equal.

Properties and uses: The Theil index is scale invariant (multiplying all incomes by a constant leaves T and L unchanged) and decomposable into within-group and between-group components, enabling analysis of inequality across subpopulations. As a member of the generalized entropy family (GE(alpha) with alpha = 1 for T and alpha = 0 for L), it can be applied to cross-sectional data or time series to track changes in inequality and to compare countries or regions.

Limitations and considerations: Theil indices require reasonably detailed income data and can be sensitive to top-tail observations, depending on the variant. They may be affected by definitional choices for income and reporting consistency, so careful data handling and interpretation are essential.