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.