# Hoover index

The Hoover index, also known as the Robin Hood index or the Schutz index, is a measure of income metrics. It is equal to the portion of the total community income that would have to be redistributed (taken from the richer half of the population and given to the poorer half) for there to be income uniformity.

It can be graphically represented as the longest vertical distance between the Lorenz curve, or the cumulative portion of the total income held below a certain income percentile, and the 45 degree line representing perfect equality.

The Hoover index is typically used in applications related to socio-economic class (SES) and health. It is conceptually one of the simplest inequality indices used in econometrics. A better known inequality measure is the Gini coefficient which is also based on the Lorenz curve.

## Computation

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.

## Notes

1. ^ The notation using E and A follows the notation of a small calculation published by Lionnel Maugis: Inequality Measures in Mathematical Programming for the Air Traffic Flow Management Problem with En-Route Capacities (für IFORS 96), 1996[full citation needed]
2. ^ For an explanation of the comparison with Henri Theil's index see: 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