Theory:

Let us look at an example to find standard deviation of a grouped data by assumed mean method.
Example:
Calculate the standard deviation of the following observations using assumed mean method.
 
\(x\)
\(1\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(f\)
\(19\)
\(5\)
\(7\)
\(23\)
\(16\)
\(13\)
 
Explanation:
 
Let the assumed mean \(A\) \(=\) \(3\).
 
Let us form a frequency distribution table.
 
\(x_{i}\)
\(f_{i}\)
\(d_{i} = x_{i} - A\)
 
\(=\) \(x_{i} - 3\)
\(f_{i} d_{i}\)
\(d_{i}^{2}\)
\(f_{i}d_{i}^{2}\)
\(1\)
\(19\)
\(-2\)
\(-38\)
\(4\)
\(76\)
\(2\)
\(5\)
\(-1\)
\(-5\)
\(1\)
\(5\)
\(3\)
\(7\)
\(0\)
\(0\)
\(0\)
\(0\)
\(4\)
\(23\)
\(1\)
\(23\)
\(1\)
\(23\)
\(5\)
\(16\)
\(2\)
\(32\)
\(4\)
\(64\)
\(6\)
\(13\)
\(3\)
\(39\)
\(9\)
\(117\)
 
\(\sum_{i = 1}^{6} f_{i} = 83\)
 
\(\sum_{i = 1}^{6}  f_{i} d_{i}\) \(=\) \(51\)
 
\(\sum_{i = 1}^{6}  f_{i} d_{i}^{2} = 285\)
The  formula to calculate the standard deviation by assumed mean method is given by:
 
\(\sigma = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right)^2}\) where \(N = \sum_{i = 1}^{n} f_{i}\).
Substitute the required values in the above formula.
 
\(\sigma = \sqrt{\frac{285}{83}- \left(\frac{51}{83}\right)^2}\)
 
\(=\) \(\sqrt{3.434 - \left(0.614 \right)^2}\)
 
\(=\) \(\sqrt{3.434 - 0.378}\)
 
\(=\) \(\sqrt{3.056}\)
 
\(=\) \(1.748\)
 
\(\approx\) \(1.75\)
 
Therefore, the standard deviation of the given data is \(1.75\).