Theory:

Let us consider the steps for finding the mean of grouped data using the step deviation method.
 
Steps:
 
1. Calculate the midpoint of the class interval and name it as \(x_i\).
 
2. From the data of \(x_i\), choose any value(preferably in the middle) as the assumed mean(\(a\)).
 
3. Determine the deviation (\(d = x_i - a\)) for each class.
 
4. Determine the deviation (\(u = \frac{x_i - a}{h}\) where \(h\) is the class size) for each class.
 
5. Multiply the frequency and \(u_i\) of each class interval and name it as \(f_iu_i\).
 
6. Calculate the mean by applying the formula \(\overline X = a + \left[\frac{\sum fd}{\sum f} \times h \right]\).
Example:
Find the mean of the following frequency distribution:
 
Class interval\(30 - 40\)\(40 - 50\)\(50 - 60\)\(60 - 70\)\(70 - 80\)
Frequency\(124\)\(156\)\(200\)\(10\)\(10\)
 
Solution:
 
Let the assumed mean be \(a = 55\) and class width \(h = 10\).
 
Class interval
Frequency
\(f_i\)
Midpoint
\(x_i\)
Deviation
\(d_i = x_i - 55\)
\(u_i = \frac{x - a}{h}\)
\(f_iu_i\)
\(30 - 40\)\(124\)\(35\)\(-20\)\(-2\)\(-248\)
\(40 - 50\)\(156\)\(45\)\(-10\)\(-1\)\(-156\)
\(50 - 60\)\(200\)\(55\)\(0\)\(0\)\(0\)
\(60 - 70\)\(10\)\(65\)\(10\)\(1\)\(10\)
\(70 - 80\)\(10\)\(75\)\(20\)\(2\)\(20\)
Total\(\sum f_i = 500\)   \(\sum f_iu_i = -374\)
We know that the mean of the grouped frequency distribution using the step deviation method can be determined using the formula, \(\overline X = a + \left[\frac{\sum f_iu_i}{\sum f_i} \times h \right]\).
Substituting the known values in the above formula, we have:
 
\(\overline X = 55 + \left[\frac{-374}{500} \times 10 \right]\)
 
\(\overline X = 55 - 7.48\)
 
\(\overline X = 47.52\)
 
Therefore, the mean of the given data is \(47.52\).