Theil Index - Formulas

Formulas

The basic Theil index, which has higher resolution for changes to higher incomes, is

T_T=T_{\alpha=1}=\frac{1}{N}\sum_{i=1}^N \left( \frac{x_i}{\overline{x}} \cdot \ln{\frac{x_i}{\overline{x}}} \right)

where is income/person. When is inverted to be people/income, or if changes in lower incomes are more important, a different formula is used that is derivable from by

T_L=T_{\alpha=0}=MLD=\frac{1}{N}\sum_{i=1}^N \left( \ln{\frac{\overline{x}}{x_i}} \right)

is also known as the MLD (mean log deviation) because it gives the mean deviation of . Sometimes the average of and is used, which has the advantage of being "symmetric" like the Gini, Hoover, and Coulter indices. "Symmetric" means it gives the same result for x as it does for 1/x:

T_S=(T_T+T_L)/2=\frac{1}{2 N}\sum_{i=1}^N \left

For these equations, is the income of the th person or subgroup, is the mean income of the persons or subgroups, and is the population or number of subgroups.

If everyone has the same income, the indices give 0 which, counter-intuitively, is when the population's income has maximum disorder. If one person has all the income, then TT gives the result, which is maximum order. Dividing TT by can normalize the equation to range from 0 to 1.

The indices measure an entropic "distance" the population is away from the "ideal" egalitarian state of everyone having the same income. The numerical result is in terms of negative entropy so that a higher number indicates more order that is further away from the "ideal" of maximum disorder. Formulating the index to represent negative entropy instead of entropy allows it to be a measure of inequality rather than equality.

If applies to the distribution of income in people, then can be used to get the same numerical result for the distribution of people in income.

The two Theil indices and are special cases of the generalized entropy index with and . The Atkinson index with is a transformation of by A=1-e^-T.

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