Theory:

Let us look at an example to find standard deviation of ungrouped data by step deviation method.
Example:
The wages of six co-workers are given below.
 
\(250\), \(260\), \(270\), \(300\), \(310\), \(330\)
 
Find its standard deviation by step deviation method.
 
Explanation:
 
Let \(n\) represent the number of co-workers.
 
\(n\) \(=\) \(6\)
 
Let \(A\) be the assumed mean, which is the middle most value.
 
Here, \(A\) \(=\) \(270\).
 
Let \(c\) be the common divisor.
 
Here, \(c = 10\).
 
Let \(x_{i}\) represent the wages of each worker.
 
\(x_{i}\)
\(x_{i} - A\)
 
\(=\) \(x_{i} - 270\)
\(d_{i} = \frac{x_{i} - A}{c}\)
 
\(=\) \(\frac{x_{i} - A}{10}\)
\(d_{i}^{2}\)
\(250\)
\(-20\)
\(-2\)
\(4\)
\(260\)
\(-10\)
\(-1\)
\(1\)
\(270\)
\(0\)
\(0\)
\(0\)
\(300\)
\(30\)
\(3\)
\(9\)
\(310\)
\(40\)
\(4\)
\(16\)
\(330\)
\(60\)
\(6\)
\(36\)
 
 
\(\sum d_{i} = 10\)
\(\sum d_{i}^{2} = 66\)
The  formula to calculate the standard deviation by step deviation method is given by:
 
\(\sigma = c \times \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\) where \(d_{i} = \frac{x_{i} - A}{c}\).
Substitute the known values in the above formula.
 
\(\sigma = 10 \times \sqrt{\frac{66}{6}- \left(\frac{10}{6}\right)^2}\)
 
\(=\) \(10 \times \sqrt{11 - (1.667)^2}\)
 
\(=\) \(10 \times \sqrt{11 - 2.779}\)
 
\(=\) \(10 \times \sqrt{8.221}\)
 
\(=\) \(10 \times 2.8672\)
 
\(=\) \(28.672\)
 
\(\approx\) \(28.67\)
 
Therefore, the standard deviation of the given data is \(28.67\).