Range the Absolute Measure of Dispersion


Statistics / Saturday, September 15th, 2018

Definition

Range is the simplest method of studying variation. It represents the differences between the values of the extremes. It is the difference between the highest and the lowest values in the series. ‘R’ is used to denote it. In frequency distribution, range is the difference between the upper limit of the largest class interval and the lower limit of the smallest class interval.
Range can be computed using following equation:
\[\text{Range }=\text{ Large Value }-\text{ Small Value}\]
\[\text{Coefficient of Range}=\frac{\text{Large Value }-\text{ Small Value}}{\text{Large Value + Small Value}}\]

In the following three articles in Statistics we will discuss various types of Absolute Measure of Dispersion in details:
1. Quartile Deviations (Q.D.)
2. Mean Deviations (M.D.)
3. Standard Deviations (S.D.)

Merits

1. It is simple and easy to understand.
2. It is rigidly defined.

Demerits

1. Affected by extreme values.
2. Not based on all values only uses extreme values.

Uses

1. It is used in Statistical Quality Control.
2. Used when deep analysis is not required.
3. It is used when data has no abnormal values.

Note: In the open class intervals Range is not defined.

 Example 01

Find the range of the following discrete series 26, 28, 28, 26, 28, 30, 27, 29, 26, 24.

Solution:
As we know that,
\[\text{Range }=\text{ Large Value }-\text{ Small Value}\]
\[\Rightarrow R=30-24=6\]
Therefore, the Range of the given discrete series is 6.

 Example 02

Find the range for the continuous series of data shown in table.

Class Interval0-55-1010-1515-2020-25
Frequency101525128

Solution:
As we know that,
\[\text{Range }=\text{ Large Value }-\text{ Small Value}\]
\[\Rightarrow R=25-0=25\]
Therefore, the range of the given continuous series is 25.

 Example 03

Compute the range and also the co-efficient of range of the given series of state which on is more dispersed and which is more uniform.
Series 1: 9, 10, 15, 19, 21
Series 2: 1, 15, 24, 28, 29

Solution:
As we know that,
\[\text{Range }=\text{ Large Value }-\text{ Small Value}\]
\[\text{Coefficient of Range}=\frac{\text{Large Value }-\text{ Small Value}}{\text{Large Value + Small Value}}\]
For Series 1
\[R=21-9=12\]
\[\text{Coefficient of Range}=\frac{21-9}{21+9}=\frac{12}{30}=0.4\]

For Series 1
\[R=29-1=28\]
\[\text{Coefficient of Range}=\frac{29-1}{29+1}=\frac{28}{30}=0.933\]

 Example 04

Find range and Co-efficient of range from the following data
Series 1: 10, 11, 12, 13, 14
Series 2: 40, 41, 42, 43, 44
Series 3: 100, 101, 102, 103, 104

Solution:
As we know that,
\[\text{Range }=\text{ Large Value }-\text{ Small Value}\]
\[\text{Coefficient of Range}=\frac{\text{Large Value }-\text{ Small Value}}{\text{Large Value + Small Value}}\]
For Series 1
\[R=14-10=4\]
\[\text{Coefficient of Range}=\frac{14-10}{14+10}=\frac{4}{24}=0.166\]

For Series 2
\[R=44-40=4\]
\[\text{Coefficient of Range}=\frac{44-40}{44+40}=\frac{4}{84}=0.0476\]

For Series 3
\[R=104-100=4\]
\[\text{Coefficient of Range}=\frac{104-100}{104+100}=\frac{4}{204}=0.0196\]
Hence, Series 1 is more dispersed and less uniform and Series 3 is less dispersed and more uniform.

 

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