Hoover index

Let ${\displaystyle x_{i}}$ be the income of the ${\displaystyle i}$-th person and ${\displaystyle {\bar {x}}}$ be the mean income. Then the Hoover index ${\displaystyle H}$ is:

${\displaystyle H={\frac {1}{2}}{\frac {\sum _{i}|x_{i}-{\bar {x}}|}{\sum _{i}x_{i}}}.}$

This value can also be computed using quantiles. For the formula, a notation[1] is used, where the amount ${\displaystyle N}$ of quantiles only appears as upper border of summations. Thus, inequities can be computed for quantiles with different widths ${\displaystyle A}$. For example, ${\displaystyle E_{i}}$ could be the income in the quantile #i and ${\displaystyle A_{i}}$ could be the amount (absolute or relative) of earners in the quantile #i. ${\displaystyle E_{\text{total}}}$ then would be the sum of incomes of all ${\displaystyle N}$ quantiles and ${\displaystyle A_{\text{total}}}$ would be the sum of the income earners in all ${\displaystyle N}$ quantiles.

Computation of the Robin Hood index ${\displaystyle H}$:

${\displaystyle H={\frac {1}{2}}\sum _{i=1}^{N}\left|{\frac {{E}_{i}}{{E}_{\text{total}}}}-{\frac {{A}_{i}}{{A}_{\text{total}}}}\right|.}$

For comparison,[2] here also the computation of the symmetrized Theil index ${\displaystyle T_{s}}$ is given:

${\displaystyle T_{s}={\frac {1}{2}}\sum _{i=1}^{N}\ln {\frac {{E}_{i}}{{A}_{i}}}\left({\frac {{E}_{i}}{{E}_{\text{total}}}}-{\frac {{A}_{i}}{{A}_{\text{total}}}}\right).}$

Both formulas can be used in spreadsheet computations.

• Atkinson index
• Diversity index
• Generalized entropy index
• Gini index
• Income inequality metrics
• Suits index
• Theil index

• Edgar Malone Hoover jr. (1936) The Measurement of Industrial Localization, Review of Economics and Statistics, 18, No. 162–71
• Edgar Malone Hoover jr. (1984) An Introduction to Regional Economics, 1984, ISBN 0-07-554440-7
• Philip B. Coulter: (1989) Measuring Inequality ISBN 0-8133-7726-9 (Describes about 50 different inequality measures.)
• Robert Sapolsky (2005) Sick of Poverty, Scientific American, December 2005
• Bruce P Kennedy, Ichiro Kawachi, Deborah Prothrow-Stith (1996) Income distribution and mortality: cross sectional ecological study of the Robin Hood index in the United States, BMJ 312:1004–07

• Free Inequality Calculator: Online and downloadable scripts (Python and Lua) for Atkinson, Gini, and Hoover inequalities