## 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

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

### MERITS OF RANGE

• Easy to compute.
• It gives the maximum spread of data.
• Easy to understand.

### DEMERITS OF RANGE

• It is affected greatly by sampling fluctuations.
• It is not based on all the observations.
• It cannot be used in case of open-end distribution.