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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 \(2.06\).
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