Spreadsheet Computations
The Gini coefficient, the Hoover index and the Theil index as well as the related welfare functions can be computed together in a spreadsheet. The welfare functions serve as alternatives to the median income.
| Group | Members per Group |
Income per Group |
Income per Individual |
Relative Deviation |
Accumulated Income |
Gini | Hoover | Theil |
|---|---|---|---|---|---|---|---|---|
| 1 | A1 | E1 | Ē1 = E1/A1 | D1 = E1/ΣE - A1/ΣA | K1 = E1 | G1 = (2K1 - E1)A1 | H1 = abs(D1) | T1 = ln(Ē1)D1 |
| 2 | A2 | E2 | Ē2 = E2/A2 | D2 = E2/ΣE - A2/ΣA | K2 = E2 + K1 | G2 = (2K2 - E2)A2 | H2 = abs(D2) | T2 = ln(Ē2)D2 |
| 3 | A3 | E3 | Ē3 = E3/A3 | D3 = E3/ΣE - A3/ΣA | K3 = E3 + K2 | G3 = (2K3 - E3)A3 | H3 = abs(D3) | T3 = ln(Ē3)D3 |
| 4 | A4 | E4 | Ē4 = E4/A4 | D4 = E4/ΣE - A4/ΣA | K4 = E4 + K3 | G4 = (2K4 - E4)A4 | H4 = abs(D4) | T4 = ln(Ē4)D4 |
| Totals | ΣA | ΣE | Ē = ΣE/ΣA | ΣG | ΣH | ΣT | ||
| Inequality Measures |
Gini = 1 - ΣG/ΣA/ΣE | Hoover = ΣH / 2 | Theil = ΣT / 2 | |||||
| Welfare Function |
WG = Ē(1 - Gini) | WH = Ē(1 - Hoover) | WT = Ē(1 - Theil) |
In the table, fields with a yellow background are used for data input. From these data inequality measures as well as the related welfare functions are computed and displayed in fields with green background.
In the example given here, "Theil index" stands for the arithmetic mean of a Theil index computed for the distribution of income within a society to the individuals (or households) in that society and a Theil index computed for the distribution of the individuals (or households) in the society to the income of that society. The difference between the Theil index and the Hoover index is the weighting of the relative deviation D. For the Hoover index the relative deviation D per group is weighted with its own sign. For the Theil index the relative deviation D per group is weighted with the information size provided by the income per individual in that group.
For the computation the society usually is divided into income groups. Often there are four or five groups consisting of a similar amount of individuals in each group. In other cases the groups are created based on income ranges which leads to having different amounts of individuals in the different groups. The table above shows a computation of inequality indices for four groups. For each group the amount of individuals (or households) per group A and the total income in that group E is specified.
The parameter pairs A and E need to be sorted for the computation of the Gini coefficient. (For the Theil index and the Hoover index no sorting is required.) A and E has the be sorted so that the values in the column "Income per individual" are lined up in ascending order.
Read more about this topic: Income Inequality Metrics
Famous quotes containing the word computations:
“That which occasions so many mistakes in the computations of men, when they expect return for favors, is that the giver’s pride and the receiver’s cannot agree upon the value of the kindness done.”
—François, Duc De La Rochefoucauld (1613–1680)