Range

Range is the simplest method of studying dispersion. It is the difference between the highest and the lowest values in a series.

$$ Range = L - S $$

where L= largest item; S = smallest item.

The relative measure corresponding to range, called the coefficient of range is obtained by applying the following formula:

$$ Coefficient \,of \,Range \,= \,{{\frac{L - S }{L + S}}} $$

Individual Series

Let us find Range and Coefficient of Range. The profits of a company for the last 8 years are given below.

Table 6.2
Year	Profit (in 000 Rs)
1985	40
1986	30
1987	80
1988	100
1989	115
1990	85
1991	210
1992	230

$$ Range = L - S $$

Here L = 230; S = 30.

Range = 230 - 30 = 200

$$ Coefficient \,of \,Range \,= \,{{\frac{L - S }{L + S}}} $$ $$ = \,{{\frac{230 - 30 }{230 + 30}}} $$ $$ = \,{{\frac{200 }{260}}} $$ $$ = \,{{0.77}} $$

Discrete Series

Let us find Range and Coefficient of Range for a discrete series.

Table 6.3
Size	Frequency
5	7
10	8
15	12
20	16
25	21
30	17
35	12
40	4

In order to find Range and Coefficient of Range, we should take the highest and the lowest values of size of items.

$$ Range = L - S $$

Here L = 40; S = 5.

Range = 40 - 5 = 35

$$ Coefficient \,of \,Range \,= \,{{\frac{L - S }{L + S}}} $$ $$ = \,{{\frac{40 - 5 }{40 + 5}}} $$ $$ = \,{{\frac{35}{45}}} $$ $$ = \,{{0.78}} $$

Continuous Series

For continuous series, range is calculated either by subtracting the lower limit of the lowest class from the upper limit of the highest class or by subtracting the mid-value of the lowest class from the midvalue of the highest class.

Let us find the range and coefficient of range of the following series:

Table 6.4
Daily Wage	Number of Workers
80 - 100	12
100 - 120	18
120 - 140	24
140 - 160	27
160 - 180	32
180 - 200	20

$$ Range = L - S $$

Here L = 200; S = 80.

Range = 200 - 80 = 120

$$ Coefficient \,of \,Range \,= \,{{\frac{L - S }{L + S}}} $$ $$ = \,{{\frac{200 - 80 }{200 + 80}}} $$ $$ = \,{{\frac{120}{280}}} $$ $$ = \,{{0.43}} $$

Let us find the range and coefficient of range of the following series where only midpoints are given:

Table 6.5
Class midpoints	Frequency
2	3
5	5
8	6
11	8
14	6
17	4
20	1

$$ Range = L - S $$

Here L = 20; S = 2.

Range = 20 - 2 = 18

$$ Coefficient \,of \,Range \,= \,{{\frac{L - S }{L + S}}} $$ $$ = \,{{\frac{20 - 2 }{20 + 2}}} $$ $$ = \,{{\frac{18}{22}}} $$ $$ = \,{{0.82}} $$