Theory:

Let us look at an example to find standard deviation of ungrouped data by direct method.
Example:
Find the standard deviation of the data \(5\), \(8\), \(10\), \(11\) and \(9\) by direct method.
 
Explanation:
 
Let \(n\) represent the number of values in the given data.
 
\(n\) \(=\) \(5\)
 
Let \(x_{i}\) represent the each value of the data.
 
\(x_{i}\)
\(x_{i}^{2}\)
\(5\)
\(25\)
\(8\)
\(64\)
\(10\)
\(100\)
\(11\)
\(121\)
\(9\)
\(81\)
\(\sum x_{i} = 43\)
\(\sum x_{i}^{2} = 391\)
The formula to calculate the standard deviation by direct method is given by:
 
\(\sigma = \sqrt{\frac{\sum x_{i}^{2}}{n}- \left(\frac{\sum x_{i}}{n}\right)^2}\)
Substitute the known values in the above formula.
 
\(\sigma = \sqrt{\frac{391}{5}- \left(\frac{43}{5}\right)^2}\)
 
\(=\) \(\sqrt{78.2 - (8.6)^2}\)
 
\(=\) \(\sqrt{78.2 - 73.96}\)
 
\(=\) \(\sqrt{4.24}\)
 
\(=\) \(2.0591\)
 
\(\approx\) \(2.06\)
 
Therefore, the standard deviation of the given data is \(2.06\).