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

Find class midpoints.

Cumulate the class midpoints .

Cumulate the frequencies.

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

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

Now mark cumulative percentages of frequencies on the xaxis and cumulative
class midpoints on the yaxis. Note that each axis will have values from 0 to 100.

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

Then plot the cumulative values and cumulative frequencies, and join these points
to get a curve.
Table 6.43  

Income  Number of persons 
0  5000  5 
5000  10000  10 
10000  20000  18 
20000  40000  10 
40000  50000  7 
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  \(\mathbf{ {{{\frac{100}{50}} \,× \,50 }\, =\,100}} \) 
(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.