## Notes in English. ## Lorenz Curve

Dispersion can be studied graphically also. For that we use what is called Lorenz Curve, after the name of Dr. Max O. Lorenz who first studied the dispersion of distribution of wealth by graphic method. This method is most commonly used to show inequality of income or wealth in a country and sometimes to make comparisons between countries or between different time periods. The Curve uses the information expressed in a cumulative manner to indicate the degree of variability. It is especially useful in comparing the variability of two or more distributions.

It has a draw back that it does not give any numerical value of the measure of dispersion. It merely gives a picture of the extent to which a series is pulled away from an equal distribution.

STEPS

1. Find class midpoints.

2. Cumulate the class midpoints .

3. Cumulate the frequencies.

4. Take the grand total of class midpoints and grand total of frequencies as 100.

5. Then convert all the other cumulative class midpoints and cumulative frequencies into their respective percentages.

6. Now mark cumulative percentages of frequencies on the x-axis and cumulative class midpoints on the y-axis. Note that each axis will have values from 0 to 100.

7. Draw a line from the origin to the point whose co-ordinate is (100, 100). This line is xcalled the line of equal distribution.

8. Then plot the cumulative values and cumulative frequencies, and join these points to get a curve.

• Let us draw a Lorenz Curve for the following data and show the inequality in income.
• Table 6.43
Income Number of persons
0 - 5000 5
5000 - 10000 10
10000 - 20000 18
20000 - 40000 10
40000 - 50000 7

We need to create a table showing midpoints, cumulative midpoints, cumulative midpoints in percentages, frequency, cumulative frequency and cumulative frequency in percentage, this is shown in the below given table.

Table 6.44
Income Midpoints Cumulative midpoints Cumulative midpoints in percentages Frequency Cumulative frequency Cumulative frequency in percentages
0 - 5000 2500 2500 $$\mathbf{ {{{\frac{100}{100000}} \,× \,2500 }\, =\,2.5}}$$ 5 5 $$\mathbf{ {{{\frac{100}{50}} \,× \,5 }\, =\,10}}$$
5000 - 10000 7500 10000 $$\mathbf{ {{{\frac{100}{100000}} \,× \,10000 }\, =\,10}}$$ 10 15 $$\mathbf{ {{{\frac{100}{50}} \,× \,15 }\, =\,30}}$$
10000 - 20000 15000 25000 $$\mathbf{ {{{\frac{100}{100000}} \,× \,25000 }\, =\,25}}$$ 18 33 $$\mathbf{ {{{\frac{100}{50}} \,× \,33 }\, =\,66}}$$
20000 - 40000 30000 55000 $$\mathbf{ {{{\frac{100}{100000}} \,× \,55000 }\, =\,55}}$$ 10 43 $$\mathbf{ {{{\frac{100}{50}} \,× \,43 }\, =\,86}}$$
40000 - 50000 45000 100000 $$\mathbf{ {{{\frac{100}{100000}} \,× \,100000 }\, =\,100}}$$ 7 50 $$\mathbf{ {{{\frac{100}{50}} \,× \,50 }\, =\,100}}$$

Note that for drawing Lorenz curve we take the frequency components along the x-axis and the corresponding value components along the y-axis. From the above table, we get the co-ordinates to be plotted as:

(x, y) = (10, 2.5), (30, 10), (66, 25), (86, 55), (100, 100) From the above figure it is clear that along the line OC, the distribution of income proportionately equal; so that 5% of the income is shared by 5% of the population, 15% of the income is shared by 15% of the population, and so on. Hence we call OC as the line of equal distribution. The farther the curve OAC from this line, the greater is the variability present in the distribution, If there are two or more curves, the one which is the farthest from the line OC has the highest dispersion.