Range and Coefficient of Range

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 and ${x_0}$ denotes the minimum observation, then the range is defined as

$Range = {x_m} – {x_0}$

In the 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 products produced in factories. The quality of products is to be kept within a certain range of values.

Range is based on 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 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 a concentration of observations in the center. In group B the observations are concentrated in the extreme corners, and in group C the observations are almost equally distributed in the interval from 30 to 50. The range fails to explain differences in the three groups of data. This defect in range cannot be removed even if we calculate the coefficient of the 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

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

$Coefficient\,of\,Range = \frac{{{x_m} – {x_0}}}{{{x_m} + {x_0}}}$

Range ${x_m} – {x_0}$ is standardized by the total ${x_m} + {x_0}$

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

Set A:   10, 15, 18, 20, 20
Set B:   30, 35, 40, 45, 50

The values of the ranges and coefficients 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 there is greater dispersion in set B, but this is not true. The range of 20 in set B is for more observations and the range of 10 in set A is for fewer observations. Thus 20 and 10 cannot be compared directly. Their base is not the same. The marks in mathematics are out of 25 and the marks of English are out of 100. Thus, it makes no sense to compare 10 with 20. When we convert these two values into coefficients of range, we see that the 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:

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

Solution:

Here the largest value $= {x_m} = 1575$ and the smallest value $= {x_0} = 1380$

Range $= {x_m} – {x_0} = 1575 – 1380 = 195$

Coefficient of Range $= \frac{{{x_{ m}} – {x_0}}}{{{x_m} + {x_0}}} = \frac{{1575 – 1380}}{{1575 + 1380}} = \frac{{195}}{{2955}} = 0.066$

Example:

The following distribution gives the number 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 the largest value $= {x_m} = 10$ and the 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.

 Weight (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:

 Weight (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   ${x_m} =$ the upper class boundary of the highest class $= 74.5$
${x_0} =$ and the lower class boundary of the lowest class $= 59.5$

Range $= {x_m} – {x_0} = 74.5 – 59.5 = 15$ Kilograms

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   ${x_m} =$ the mid value of the highest class $= 73$
${x_0} =$ and the mid value of the lowest class $= 61$

Range $= {x_m} – {x_0} = 73 – 61 = 12$ Kilograms

Coefficient of Range $= \frac{{{x_{m}} – {x_0}}}{{{x_m} + {x_0}}} = \frac{{73 – 61}}{{73 + 61}} = \frac{{12}}{{134}} = 0.0895$