Ugrás a tartalomhoz

## SOCIAL STATISTICS

Renáta Németh, Dávid Simon

ELTE

Special measures of variability

## Special measures of variability

### Decile ratio

Compared to the range, less sensitive to outliers.

Appropriate for interval-ratio variables.

Decile ratio = mean income of top 10 percent / mean income of bottom 10 percent.

Compared to the IQR, it concerns the distance between the two ends of the distribution. Therefore it is commonly used for measuring income inequalities (distance between the richest and the poorest).

Example for its calculation

Consider the sample of size 30 below, ordered by income (fictive numbers):

 1 42,720 2 43,866 3 45,821 4 49,418 5 49,781 6 50,975 7 53,739 8 57,693 9 69,131 10 89,341 11 111,940 12 137,045 13 150,307 14 156,443 15 156,498 16 208,115 17 227,996 18 235,034 19 249,609 20 262,369 21 300,046 22 328,424 23 348,137 24 351,597 25 362,036 26 368,305 27 372,850 28 447,664 29 449,088 30 484,355

Mean within the top 10 percent is (42,720+43,866+45,821)/3=44,802, while the mean within the bottom 10 percent is (447,664+449,088+484,355)/3=460,369. Hence the decile ratio is 460,369/44,802=10.3.

Example

Many researches claim that income inequalities increased during the transition period in Hungary, until about 1995. Data below support this statement.

(source: Társadalmi helyzetkép 2002, Central Bureau of Statistics). Interpret the figure.

### Gini coefficient, Lorenz curve

The Gini coefficient is a commonly used measure of inequality of income or wealth, mainly in economic areas as health-economy or economic sociology.

Contrary to the IQR or the decile ratio, it takes into account the whole distribution.

As we have seen in Section Time series chart, the Gini can range from 0 to 1. A value of 0 expresses total equality and a value of 1 total inequality. A value of 0.4 can be interpreted as a rather high inequality.

The Gini coefficient is usually defined mathematically by the Lorenz curve, which itself is (a more complex) measure of inequality. The Lorenz curve plots

• the proportion of the total income of the population (y axis) that is cumulatively earned

• by the bottom x% of the population (x axis).

The line at 45 degrees represents perfect equality: Points of the curve correspond to conclusions like "The poorest 60% of adults earns only 40% of the total income.

The Gini coefficient is defined as twice the area lying between the line of equality and the Lorenz curve. The Gini computed from the above curve is 0.31.

(Source of data: Hungarian National Health Survey 2000. Income is defined as per capita household net income.)

Case study – income differences in Hungary

In 2000, the Gini had a value of 0.31 in Hungary.

Compare: in the 90’s, Gini ranged from approximately 0.25 (Eastern-European countries) to 0.5 (Latin-America) although not every country has been assessed.

What may affect income inequalities?

Figures below show that the Gini in Hungary tends to be higher

• among higher educated jobs, and   