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Let us look at an example to find standard deviation of a grouped data by assumed mean method.
Example:
Calculate the standard deviation of the following observations using assumed mean method.
| \(x\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) |
| \(f\) | \(19\) | \(5\) | \(7\) | \(23\) | \(16\) | \(13\) |
Explanation:
Let the assumed mean \(A\) \(=\) \(3\).
Let us form a frequency distribution table.
\(x_{i}\) | \(f_{i}\) | \(d_{i} = x_{i} - A\) \(=\) \(x_{i} - 3\) | \(f_{i} d_{i}\) | \(d_{i}^{2}\) | \(f_{i}d_{i}^{2}\) |
\(1\) | \(19\) | \(-2\) | \(-38\) | \(4\) | \(76\) |
\(2\) | \(5\) | \(-1\) | \(-5\) | \(1\) | \(5\) |
\(3\) | \(7\) | \(0\) | \(0\) | \(0\) | \(0\) |
\(4\) | \(23\) | \(1\) | \(23\) | \(1\) | \(23\) |
\(5\) | \(16\) | \(2\) | \(32\) | \(4\) | \(64\) |
\(6\) | \(13\) | \(3\) | \(39\) | \(9\) | \(117\) |
\(\sum_{i = 1}^{6} f_{i} = 83\) | \(\sum_{i = 1}^{6} f_{i} d_{i}\) \(=\) \(51\) | \(\sum_{i = 1}^{6} f_{i} d_{i}^{2} = 285\) |
The formula to calculate the standard deviation by assumed mean method is given by:
\(\sigma = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right)^2}\) where \(N = \sum_{i = 1}^{n} f_{i}\).
Substitute the required values in the above formula.
\(\sigma = \sqrt{\frac{285}{83}- \left(\frac{51}{83}\right)^2}\)
\(=\) \(\sqrt{3.434 - \left(0.614 \right)^2}\)
\(=\) \(\sqrt{3.434 - 0.378}\)
\(=\) \(\sqrt{3.056}\)
\(=\) \(1.748\)
\(\approx\) \(1.75\)
Therefore, the standard deviation of the given data is \(1.75\).
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