Hoover index
The Hoover inequality is the most direct and simplest of all the unequal distribution coefficients. It describes the relative deviation from the mean. It is " right " because they, for example, in an unequal distribution of money simply describes the proportion of the total money that would have to be redistributed in order to make a difference in distribution is a uniform distribution. Other names for the Hoover inequality are Hoover coefficient, Hoover index, Balassa -Hoover index, Hoover concentration index and segregation and dissimilarity index.
Calculation example
The Hoover unequal distribution can also - like the Gini coefficient - for income distributions, calculate for asset distributions and other distributions. How to calculate the Hoover inequality, the following example shows based on the distribution of a " total assets " of about 10 trillion Deutsche Mark in Germany (1995 ):
50 percent of the population (A1 ) had 2.5 percent of the assets (E1). 40 percent of the population (A2 ) had 47.5 percent of the assets (E2). 9 percent of the population (A3 ) had 27.0 percent of the assets ( E3). 1 percent of the population (A4 ) had 23.0 percent of the assets (E4 ). In a first step the data is "normalized" shown ( Etotal = anet = 1):
A1 = 0.50 E1 = 0.025 A2 = 0.40 E2 = 0.475 A3 = 0.09 E3 = 0.270 A4 = 0.01 E4 = 0.230 In the second step, the absolute differences are summed up:
Abs ( E1 - A1 ) = 0.475 abs (A2 - A2 ) = 0.075 abs ( E3 - A3 ) = 0.180 abs ( E4 - A4 ) = 0.220 Sum = 0.950 Half of the sum is the Hoover inequality:
Hoover inequality: sum / 2 = 0.475 = 47.5 % Other inequality measures " interpret " unequal distributions. An example are some Entropiemaße (eg by Theil, Atkinson, Kullback Leibler and etc.) which make reference to uniform distributions of state variables in statistical physics. The Hoover coefficient is, however, very easy to understand and calculate. It directly describes the proportion of unevenly distributed resource, which would have to be redistributed, an equal distribution of this resource should be achieved. In the example would therefore 47.5 % of the assets must be redistributed if all the same amount would have to possess. ( The unequal distribution within the four defined by quantiles with different distance ranges with different width, however, it would have been ignored. )
The range of this relative inequality measure is between 0 and 1 (or between 0% and 100 %). The Hoover inequality is in a group of measures of concentration.
Formula
The complete formula of the Hoover inequality is:
In the formula, a notation is employed in which the number of quantile with defined ( with the same or different spacing ) areas appear ( with identical or different width), in the formulas, only the upper limit of the summation. This also unequal distributions can be calculated for which the areas have a different width: whether the income in the ith area and is the number ( or percentage ) of income earners in the ith area. is the sum of the incomes of all N regions and is the sum of income earners of all N regions (or 100 %). ( Of course, other allocations are also possible: for example, may also represent or property is one kind of molecules in a mixture, and for a different kind of molecules.. )
In the Hoover inequality, the individual deviations from the parity only with their own sign (ie a factor of 1 or -1) are weighted. For comparison, consider the symmetrized Theil index. In the Theil index, the individual deviations from parity are weighted based on its own information content: