Theory:

Let us look at an example to find standard deviation of ungrouped data by mean method.
Example:
Find the standard deviation of the data \(5.4\), \(8.9\), \(10.1\), \(11.4\) and \(9.2\) by mean method.
  
Explanation:
 
Let \(n\) represent the number of values in the data.
 
\(n\) \(=\) \(5\)
 
Let \(\overline x\) represent the mean of the given data.
Mean \(\overline x = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}\)
\(\overline x\) \(=\) \(\frac{5.4 + 8.9 + 10.1 + 11.4 + 9.2}{5}\)
 
\(=\) \(\frac{45}{5}\)
 
\(=\) \(9\)
 
Let \(x_{i}\) represent each values in the given data.
 
\(x_{i}\)
\(d_{i} = x_{i} - \overline x\)
 
\(=\) \(x_{i} - 9\)
\(d_{i}^{2}\)
\(5.4\)
\(-3.6\)
\(12.96\)
\(8.9\)
\(-0.1\)
\(0.01\)
\(10.1\)
\(1.1\)
\(1.21\)
\(11.4\)
\(2.4\)
\(5.76\)
\(9.2\)
\(0.2\)
\(0.04\)
 
 
\(\sum d_{i}^{2} = 19.98\)
The formula to calculate the standard deviation by mean method is given by:
 
\(\sigma\) \(=\) \(\sqrt{\frac{\sum d_{i}^{2}}{n}}\) where \(d_{i} = x_{i} - \overline{x}\)
Substitute the known values in the above formula.
 
\(\sigma\) \(=\) \(\sqrt{\frac{19.98}{5}}\)
 
\(=\) \(\sqrt{3.996}\)
 
\(=\) \(1.999\)
 
\(\approx\) \(2\)
 
Therefore, the standard deviation for the given data is \(2\).