Theory:

Consider the frequency distribution table.
 
Height (in m)\(30 - 40\)\(40 - 50\)\(50 - 60\)\(60 - 70\)\(70 - 80\)
Number of trees\(124\)\(156\)\(200\)\(10\)\(10\)
 
The above frequency table shows that the data are grouped in class intervals. This table shows that the number of trees with various heights.
 
Consider the interval \(50 - 60\). There are \(200\) trees in the heights between \(50 - 60\) metres. In grouped frequency, the individual observations are not available. Thus, we need to determine the value that indicates the particular interval. This value is called a midpoint or class mark. The midpoint can be determined using the formula:
 
Midpoint \(= \frac{UCL + LCL}{2}\)
 
Where \(UCL\) is the upper class limit and \(LCL\) is the lower class limit.
Example:
Consider the interval \(40 - 50\). Let us find the midpoint of this interval.
 
Here, \(UCL = 40\) and \(LCL = 50\)
 
Midpoint of \(40 - 50\) is \(\frac{40 + 50}{2} =\) \(\frac{90}{2}\) \(= 45\)
 
Therefore, the midpoint of the interval \(40 - 50\) is \(45\).
The arithmetic mean of a grouped frequency distribution can be determined using any one of the following methods.
  • Direct method
  • Assumed mean method
  • Step deviation method