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 more educated adults,

  • among higher educated jobs, and

  • among younger adults.

The effect of age is the strongest (Gini among young people is double of that among elderly people).

(Source: Hungarian National Health Survey 2000)