Theil Index Articles about
Theil 		 

Information About

Theil Index
APPAREL
BABY
BEAUTY
BOOKS
CAR TOYS
CELL PHONES
DVD'S
ELECTRONICS
GOURMET FOOD
GROCERIES
HEALTH & PERSONAL
HOME & GARDEN
JEWELRY
MUSIC
MUSIC INSTRUMENTS
OFFICE PRODUCTS
SOFTWARE
SPORTING GOODS
TOOLS & HARDWARE
TOYS
VIDEO GAMES
SHOPPING HOME
MORE SHOPPING...




MATHEMATICS


The formula is


T=\sum_{i=1}^N \left( rac{x_i}{\sum_{j=1}^N x_j} \cdot \ln{ rac{x_i}{\overline{x}}} ight)


where x_i is the income of the ith person, \overline{x} is the mean income, and N is the number of people. The first term inside the sum can be considered the individual's share of aggregate income, and the second term is that person's income relative to the mean. If everyone has the same (i.e., mean) income, then the index = 0. If one person has all the income, then the index = \ln{N}.

The Theil index is derived from Shannon 's measure of Information Entropy . Letting T be the Theil Index and S be Shannon's measure,


T=\ln(N)-S


Shannon derived his entropy measure in terms of the Probability of an event occurring. This can be interpreted in the Theil as the probability a dollar drawn at random from the population came from a specific individual. This is the same as the first term, the individual's share of aggregate income.


DECOMPOSABILITY

One of the advantages of the Theil index is that it is the weighted sum of inequality within subgroups. For example, inequality within the United States is the sum of each state's inequality weighted by the state's income relative to the entire country.

If the population is divided into m certain subgroups and s^k is the income share of group k, T^k is the Theil index for that subgroup, and \overline{x}^k is the average income in group k, then the Theil index is


T = \sum_{k=1}^m s^k T_k + \sum_{k=1}^m s^k \ln{ rac{\overline{x}^k}{\overline{x}}}


Therefore, one can say that a certain group "contributes" a certain amount of inequality to the whole.

Another, more popular, measure of inequality is the Gini Coefficient . The Gini coefficient is more intuitive to many people since it is based on the Lorenz Curve . However, it is not easily decomposable like the Theil.


REFERENCES

Introduction to the Theil index from the University of Texas



Articles about Theil Index
Theil-sur-vanne Theil Index
Theiler (crater) Theilheim