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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.