Theory:

The mode of the grouped frequency distribution can be determined using the formula:
 
Mode \(= l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
 
The class interval with maximum frequency is called the modal class.
 
Where \(l\) is the lower limit of the modal class,
 
\(f_1\) is the frequency of the modal class,
 
\(f_0\) is the frequency of the class preceding the modal class,
 
\(f_2\) is the frequency of the class succeeding the modal class, and
 
\(h\) is the width of the class interval.
Example:
Find the mode of the following data:
 
Class interval\(130 - 140\)\(140 - 150\)\(150 - 160\)\(160 - 170\)\(170 - 180\)
Frequency\(5\)\(36\)\(14\)\(28\)\(1\)
 
Solution:
 
The maximum frequency is \(36\), and the modal class is \(140 - 150\).
The mode of the grouped frequency distribution can be determined using the formula:
 
Mode \(= l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h\)
Here, \(l = 140\), \(f_1 = 36\), \(f_0 = 5\), \(f_2 = 14\), \(h = 10\)
 
Substituting the known values in the above formula, we have;
 
Mode \(= 140 + \left(\frac{36 - 5}{2(36) - 5 - 14} \right) \times 10\)
 
\(= 140 + \left(\frac{36 - 5}{72 - 5 - 14} \right) \times 10\)
 
\(= 140 + (\frac{31}{53}) \times 10\)
 
\(= 140 + 0.585 \times 10\)
 
\(= 140 + 5.85\)
 
\(= 145.85\)
 
Therefore, the mode of the given data is \(145.85\).