Theil index

The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. 'The best-known entropy measures are Theil’s T ( T T {displaystyle T_{T}} ) and Theil’s L ( T L {displaystyle T_{L}} ), both of which allow one to decompose inequality into the part that is due to inequality within areas (e.g. urban, rural) and the part that is due to differences between areas (e.g. the rural-urban income gap). Typically at least three-quarters of inequality in a country is due to within-group inequality, and the remaining quarter to between-group differences.' The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. The Theil index TT is the same as redundancy in information theory which is the maximum possible entropy of the data minus the observed entropy. It is a special case of the generalized entropy index. It can be viewed as a measure of redundancy, lack of diversity, isolation, segregation, inequality, non-randomness, and compressibility. It was proposed by econometrician Henri Theil at the Erasmus University Rotterdam. For a population of N 'agents' each with characteristic x, the situation may be represented by the list xi (i = 1,...,N) where xi is the characteristic of agent i. For example, if the characteristic is income, then xi is the income of agent i. The Theil T index is defined as and the Theil L index is defined as where μ {displaystyle mu } is the mean income: Equivalently, if the situation is characterized by a discrete distribution function fk (k = 0,...,W) where fk is the fraction of the population with income k and W = Nμ is the total income, then ∑ k = 0 W f k = 1 {displaystyle sum _{k=0}^{W}f_{k}=1} and the Theil index is: where μ {displaystyle mu } is again the mean income: Note that in this case income k is an integer and k=1 represents the smallest increment of income possible (e.g., cents).