Calculation of Range and Coefficient of Range

What is Range?

Range is the easiest to understand of all the measures of dispersion. The difference between the largest and smallest item in a distribution is called range. It can be written as:

Range (R) = Largest item (L) – Smallest item (S)

For example, If the marks of 5 students of class XIth are 20, 15, 18, 14, 17. After arranging the marks in ascending (or descending) order, the difference between the highest and lowest scores will be the range of marks. Therefore, the range in this case will be 20 – 14 = 6.

The range, which is expressed in the units of measurement of the given data, is an absolute measure of dispersion. A larger range value indicates greater dispersion, while a smaller range value shows lesser dispersion. The range will have a value of 0 if all the items are the same, indicating no dispersion between the items.

What is Coefficient of Range?

The ratio of the difference between two extreme items (the largest and smallest) of the distribution to their sum is known as the Coefficient of Range. The coefficient of the range is a relative measure of dispersion. Symbolically, the range can be expressed as:

Calculation of Range and Coefficient of Range

Range and Coefficient of Range can be calculated in three different series:

(I) Individual Series

Example 1:

The salaries of 8 factory employees are listed below. Calculate the range and the coefficient of the range. The following salaries are in Indian rupees: 

1400, 1450, 1520, 1380, 1485, 1495, 1575, and 1440.

Solution:

In ascending order, the wages are 1380, 1400, 1440, 1450, 1485, 1495, 1520, 1575

From the given values of salary, the Largest Item (L) = 1575, and the Smallest Item (S) = 1380

Range = L- S

= 1575 – 1380

Range = 195

Coefficient of Range = 0.065

Example 2:

Find the range and coefficient of the range of the following data:

43.1, 13.6, 18.5, 38.1, 61.4, 29.3

Solution:

In ascending order, the values are: 13.6, 18.5, 29.3, 38.1, 43.1, 61.4

Here, L = Largest Value; i.e., 61.4, S = Smallest Value;i.e., 13.6

Range = L – S

= 61.4 – 13.6

Range = 47.8

Coefficient of Range = 0.64

(II) Discrete Series

The values of the largest (L) and smallest (S) items in a discrete series should not be confused with the largest and smallest frequencies. They represent the largest and smallest values of the variable. Therefore, the range is determined without taking into account their frequencies by subtracting the smallest item from the largest item.

Example 1:

The number of homes and the number of people per home are shown in the distribution below. Find the range and coefficient of the range of the following distribution:

Solution:

Range (R) = Largest Item (L) – Smallest Item (S)

= 8 -1

Range = 7

Coefficient of Range = 0.77

Example 2:

The number of workers and production per day is shown in the distribution below. Find the range and coefficient of the range of the following distribution:

Solution:

Range = L – S

Here, L = Largest Value; i.e., 250 S = Smallest Value; i.e., 150

= 250 – 150

Range = 100

Coefficient of Range = 0.25

(III) Continuous Series

There are two ways to compute the range and coefficient of range for continuous frequency distributions:

1. First Method: Calculate the difference between the lower limits of the lowest-class interval and the upper limit of the highest-class interval.

2. Second Method: Calculate the difference between the mid-points of the lowest-class interval and the highest-class interval.

Note: Both methods will provide different results. However, both answers will be accurate.

Example 1:

The following data represents the weight of students in kg. Find the range and coefficient of the range using both methods.

Solution:

Range and Coefficient of Range by the First Method:

Range (R) = Largest Item (L) – Smallest Item (S)

= 62 – 50

Range = 12

Coefficient of Range = 0.107

Range and Coefficient of Range by the Second Method:

Range (R) = Mid-point of the Highest Class – Mid-point of the Lowest Class

= 61 – 51

Range = 10

Coefficient of Range= 0.089

Example 2:

Find the range and coefficient of range of the following series:

Solution:

In the above table, an inclusive series of marks and the number of students are given. First of all, the inclusive series will be converted into the exclusive series, after that, the largest and smallest value of the exclusive series of marks will be utilised to calculate the range and coefficient of range.

Range and Coefficient of Range by the First Method:

Range = L – S

Here, L = Largest value; i.e., 49.5, S = Smallest value; i.e., 9.5

Range = 49.5 – 9.5

Range = 40

Coefficient of Range = 0.67

Range and Coefficient of Range by the Second Method:

Range (R) = Mid-points of the Highest Class – Mid-points of the Lowest Class

= 44.5 – 14.5

Range = 30

Coefficient of Range = 0.51