The Advantages of Using Group Means in Estimating the Lorenz Curve and Gini Index from Grouped Data

• Activity Number: 647
• Type: Contributed
• Date/Time: Thursday, August 7, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Business and Economic Statistics Section
• Author(s): Merritt Lyon*+ and Li Cheung and Brian Dumbacher and Joseph Gastwirth
• Companies: Compass Lexecon and George Washington University and George Washington University and George Washington University
• Keywords: Grouped Data ; Lorenz Curve ; Gini Index ; Gini Bounds

Abstract

A recent article in the American Statistician proposed a histogram-based method for estimating the Lorenz curve and Gini index from grouped data that ignored the group means typically reported by government agencies. Comparing the method with existing ones utilizing group means, the article assumed that the group mid-points were the group means. However, as the density of the income distribution tends to increase and then decrease, this approximation is questionable. After reviewing the theoretical basis for the additional information contained in the means, we shown that as the number of groups increases, the distance between the bounds on the Gini index obtained from methods using the group means decreases, while this is not necessarily true for the histogram-based method. These results are illustrated on the U.S. Census Bureau's 2010 and 2012 household income data, which is reported in 42 groups. A simple linear interpolation method incorporating the group means is also provided, which yields an estimated Gini index between the upper and lower bounds. Lastly, we show how the Gini bounds can be used to choose more informative grouping intervals for summarizing the income data.