Range and Coefficient of Range

The Range:

Range is defined as the difference between the maximum and the minimum observation of the given data. If ${x_m}$ denotes the maximum observation ${x_0}$ denotes the minimum observation then the range is defined as

In case of grouped data, the range is the difference between the upper boundary of the highest class and the lower boundary of the lowest class. It is also calculated by using the difference between the mid points of the highest class and the lowest class. It is the simplest measure of dispersion. It gives a general idea about the total spread of the observations. It does not enjoy any prominent place in statistical theory. But it has its application and utility in quality control methods which are used to maintain the quality of the products produced in factories. The quality of products is to be kept within certain range of values.

The range is based on the two extreme observations. It gives no weight to the central values of the data. It is a poor measure of dispersion and does not give a good picture of the overall spread of the observations with respect to the center of the observations. Let us consider three groups of the data which have the same range:

Group A:   30, 40, 40, 40, 40, 40, 50
Group B:   30, 30, 30, 40, 50, 50, 50
Group C:   30, 35, 40, 40, 40, 45, 50

In all the three groups the range is 50 – 30 = 20. In group A there is concentration of observations in the center. In group B the observations are friendly with the extreme corner and in group C the observations are almost equally distributed in the interval from 30 to 50. The range fails to explain these differences in the three groups of data. This defect in range cannot be removed even if we calculate the coefficient of range which is a relative measure of dispersion. If we calculate the range of a sample, we cannot draw any inferences about the range of the population.

Coefficient of Range:

It is relative measure of dispersion and is based on the value of range. It is also called range coefficient of dispersion. It is defined as:

The range ${x_m} - {x_0}$ is standardized by the total ${x_m} + {x_0}$

Let us take two sets of observations. Set A contains marks of five students in Mathematics out of 25 marks and group B contains marks of the same student in English out of 100 marks.

Set A:   10, 15, 18, 20, 20
Set B:   30, 35, 40, 45, 50
The values of range and coefficient of range are calculated as:

 Range Coefficient of Range Set A: (Mathematics) $20 - 10 = 10$ $\frac{{20 - 10}}{{20 + 10}} = 0.33$ Set B: (English) $50 - 30 = 20$ $\frac{{50 - 30}}{{50 + 30}} = 0.25$

In set A the range is 10 and in set B the range is 20. Apparently it seems as if there is greater dispersion in set B. But this is not true. The range of 20 in set B is for large observations and the range of 10 in set A is for small observations. Thus 20 and 10 cannot be compared directly. Their base is not the same. Marks in Mathematics are out of 25 and marks of English are out of 100. Thus, it makes no sense to compare 10 with 20. When we convert these two values into coefficient of range, we see that coefficient of range for set A is greater than that of set B. Thus there is greater dispersion or variation in set A. The marks of students in English are more stable than their marks in Mathematics.

Example:

Following are the wages of 8 workers of a factory. Find the range and the coefficient of range. Wages in Dollars 1400, 1450, 1520, 1380, 1485, 1495, 1575, 1440.

Solution:

Here Largest value $= {x_m} = 1575$ and Smallest Value $= {x_0} = 1380$
Range $= {x_m} - {x_0} = 1575 - 1380 = 195$
Coefficient of Range $= \frac{{{x_{ m}} - {x_0}}}{{{x_m} + {x_0}}} = \frac{{1575 - 1380}}{{1575 + 1380}} = \frac{{195}}{{2955}} = 0.66$

Example:

The following distribution gives the numbers of houses and the number of persons per house.

 Number of Persons $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ Number of Houses $26$ $113$ $120$ $95$ $60$ $42$ $21$ $14$ $5$ $4$

Calculate the range and coefficient of range.

Solution:
Here Largest value $= {x_m} = 10$ and Smallest Value $= {x_0} = 1$
Range $= {x_m} - {x_0} = 10 - 1 = 9$
Coefficient of Range $= \frac{{{x_{m}} - {x_0}}}{{{x_m} + {x_0}}} = \frac{{10 - 1}}{{10 + 1}} = \frac{9}{{11}} = 0.818$

Example:

Find the range of the weight of the students of a university.

 Weights (Kg) $60 - 62$ $63 - 65$ $66 - 68$ $69 - 71$ $72 - 74$ Number of Students $5$ $18$ $42$ $27$ $8$

Calculate the range and coefficient of range.

Solution:

 Weights (Kg) Class Boundaries Mid Value No. of Students $60 - 62$ $59.5 - 62.5$ $61$ $5$ $63 - 65$ $62.5 - 65.5$ $64$ $18$ $66 - 68$ $65.5 - 68.5$ $67$ $42$ $69 - 71$ $68.5 - 71.5$ $70$ $27$ $72 - 74$ $71.5 - 74.5$ $73$ $8$

Method 1:
Here   ${x_m} =$ Upper class boundary of the highest class $= 74.5$
${x_0} =$ Lower class boundary of the lowest class $= 59.5$
Range $= {x_m} - {x_0} = 74.5 - 59.5 = 15$ Kilogram
Coefficient of Range $= \frac{{{x_{m}} - {x_0}}}{{{x_m} + {x_0}}} = \frac{{74.5 - 59.5}}{{74.5 + 59.5}} = \frac{{15}}{{134}} = 0.1119$

Method 2:
Here   ${x_m} =$ Mid value of the highest class $= 73$
${x_0} =$ Mid value of the lowest class $= 61$
Range $= {x_m} - {x_0} = 73 - 61 = 12$ Kilogram
Coefficient of Range $= \frac{{{x_{m}} - {x_0}}}{{{x_m} + {x_0}}} = \frac{{73 - 61}}{{73 + 61}} = \frac{{12}}{{134}} = 0.0895$