The Theoretical Basics of Popular Inequality MeasuresTravis Hale, University of Texas Inequality ProjectThis document explores several inequality measures used broadly in the literature, with a special emphasis on how to compute Theils T statistic. Inequality is related to several mathematical concepts, including dispersion, skewness, and variance. As a result, there are many ways to measure inequality, which itself arises from various social and physical phenomena. While this is not an exhaustive discussion of inequality measures, it does deal with several of the most popular statistics. Several examples are included that pertain to inequality of salaries within two fictional companies Universal Widget and Worldwide Widget but all of the inequality measures discussed apply to a broad set of research questions. The salary schedules for the example problems are below, followed by discussions of range, range ratios, the McLoone Index, the coefficient of variation, and the Gini Coefficient. Following these brief introductions is an extended description of Theils T statistic.Universal Widget Salary SchedulePosition# of Employees in PositionExact Annual SalaryCustodial Staff7$ 18,000.00Office Staff10$ 22,000.00Equipment Operators280$25,000.00Equipment Technicians15$35,000.00Foremen15$40,000.00Salespersons50$ 60,000.00Engineers10$75,000.00Managers6$ 80,000.00Vice Presidents4$ 120,000.00Senior Vice Presidents2$ 200,000.00CEO1$ 1,000,000.00Worldwide Widget Salary SchedulePosition# of Employees in PositionExact Annual SalaryCustodial Staff12$15,000.00Office Staff25$ 20,000.00Equipment Operators1000$30,000.00Equipment Technicians35$35,000.00Foremen100$45,000.00Salespersons80$50,000.00Managers10$ 60,000.00Engineers25$ 80,000.00Vice Presidents8$175,000.00Senior Vice Presidents4$250,000.00CEO1$5,000,000.00RangePerhaps the simplest measure of dispersion, the range merely calculates the difference between the highest and lowest observations of a particular variable of interest. Strengths of the range include its mathematical simplicity and ease of understanding. However, it is a very limited measure. The range only uses two observations from the overall set, it does not weight observations by important underlying characteristics (like the population of a state, the experience of an employee, etc.), and it is sensitive to inflationary pressures. In the case of a company, the range between the salaries of the highest and lowest paid employees may not give much information. For Universal Widget, the range in salaries is $982,000 ($1,000,000 - $18,000), while for Worldwide Widget the range is $4,985,000 ($5,000,000 - $15,000). Does this mean that Worldwide Widget has a much more unequal wage structure than Universal Widget? Not without further evidence.Range RatiosTo find the range ratio for a certain variable, divide the value at a certain percentile (usually above the median) by the value at a lower percentile (usually below the median). One range ratio often used in the study of inequality in educational expenditures is the Federal Range Ratio, which divides the difference between the revenue for the student at the 95th percentile and the 5th percentile by the revenue for the student at the 95th percentile.1 Another popular range ratio is the inter-quartile range ratio. Subtracting the observation at the 25th percentile by the observation at the 75th percentile results in a quantity known as the inter-quartile range, and dividing the observation at the 75th percentile by the 25th percentile calculates the inter-quartile range ratio. Range ratios can measure all sorts of inequalities and the percentiles can be constructed in any manner. A range ratio can take on any value between one and infinity, and smaller values reflect lower inequality.Using the example data, one can compute a 90:10 range ratio for the two widget companies. For Universal Widget, the 90th percentile falls at a salary of $60,000 and the 10th percentile is $25,000. Thus, the 90:10 range ratio is $60,000/$25,000 or 2.4. For Worldwide Widget, the 90th percentile falls at a salary of $35,000 and the 10th percentile is $30,000. Therefore, the 90:10 range ratio is $35,000/$30,000 or 1.17. Given this information, Worldwide Widget has a more equal pay structure, the opposite conclusion gleaned from the range.Range ratios are easy to understand and simple to compute. They can directly compares the “haves” - observations at the 90th percentile or elsewhere above the median value -with the “have-nots” - observations at the 10th percentile or elsewhere below the median, without being sensitive to outliers at the very top or very bottom of the distribution. However, like the range, range ratios only look at two distinct data points, throwing away