Measures of Dispersion 

 

8.2  Measures of Dispersion 
 
Range 
 
  Formula   
     
  \(\text{Range} = \text{Largest value} - \text{Smallest value}\)  
     
 
  Example  
     
  Given a set of data \(36, 25, 15, 26, 50, 27, 20\), determine the range of this set of data   
     
 

Solution:

1. Arrange a set of data in ascending order. 

\(15,20,25,26,27,36,50\)

2. Apply \(\text{Range} = \text{Largest value} - \text{Smallest value}\)

\(\therefore 50 -15 = 35.\)

 
 
Interquartile range of set of ungrouped data 
 
  • When the values of a set of data are arranged in ascending order, the first quartile, \(Q_1\) is the value of data that is the first \(\dfrac{1}{4}\) position
  • While the third quartile, \(Q_3\) is the value of data that is the third \(\dfrac{3}{4}\) position
 
Variance and standard deviation 
 
  • Its are the measures of dispersion commonly used in statistics. 
  • The variance is the average of the square of the difference between each data and the mean. 
  • The standard deviation is the square root of variance which is also measures the dispersion of data set relative to its mean. 
 
  Formula   
     
 

Variance 

\(\sigma^2= \dfrac{\sum (x- \bar{x})}{N}\)

 
     
 

Standard deviation

\(\sigma= \sqrt{\dfrac{\sum (x- \bar{x})}{N}}\) or \(\sigma= \sqrt{\dfrac{\sum x^2}{N}-\bar{x}^2}\)

 
 
Box Plot 
 
Box plot is a way of showing the distribution of a set of data based on five values, namely the minimum value, first quartile, median, third quartile, and the maximum value of set of data. 
 
  Example of box plot   
     
   
     
 
 
 
 
 
 

Measures of Dispersion 

 

8.2  Measures of Dispersion 
 
Range 
 
  Formula   
     
  \(\text{Range} = \text{Largest value} - \text{Smallest value}\)  
     
 
  Example  
     
  Given a set of data \(36, 25, 15, 26, 50, 27, 20\), determine the range of this set of data   
     
 

Solution:

1. Arrange a set of data in ascending order. 

\(15,20,25,26,27,36,50\)

2. Apply \(\text{Range} = \text{Largest value} - \text{Smallest value}\)

\(\therefore 50 -15 = 35.\)

 
 
Interquartile range of set of ungrouped data 
 
  • When the values of a set of data are arranged in ascending order, the first quartile, \(Q_1\) is the value of data that is the first \(\dfrac{1}{4}\) position
  • While the third quartile, \(Q_3\) is the value of data that is the third \(\dfrac{3}{4}\) position
 
Variance and standard deviation 
 
  • Its are the measures of dispersion commonly used in statistics. 
  • The variance is the average of the square of the difference between each data and the mean. 
  • The standard deviation is the square root of variance which is also measures the dispersion of data set relative to its mean. 
 
  Formula   
     
 

Variance 

\(\sigma^2= \dfrac{\sum (x- \bar{x})}{N}\)

 
     
 

Standard deviation

\(\sigma= \sqrt{\dfrac{\sum (x- \bar{x})}{N}}\) or \(\sigma= \sqrt{\dfrac{\sum x^2}{N}-\bar{x}^2}\)

 
 
Box Plot 
 
Box plot is a way of showing the distribution of a set of data based on five values, namely the minimum value, first quartile, median, third quartile, and the maximum value of set of data. 
 
  Example of box plot