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Book Free DemoLet us look at an example to find standard deviation of ungrouped data by step deviation method.
Example:
The wages of six co-workers are given below.
\(250\), \(260\), \(270\), \(300\), \(310\), \(330\)
Find its standard deviation by step deviation method.
Explanation:
Let \(n\) represent the number of co-workers.
\(n\) \(=\) \(6\)
Let \(A\) be the assumed mean, which is the middle most value.
Here, \(A\) \(=\) \(270\).
Let \(c\) be the common divisor.
Here, \(c = 10\).
Let \(x_{i}\) represent the wages of each worker.
\(x_{i}\) | \(x_{i} - A\) \(=\) \(x_{i} - 270\) | \(d_{i} = \frac{x_{i} - A}{c}\) \(=\) \(\frac{x_{i} - A}{10}\) | \(d_{i}^{2}\) |
\(250\) | \(-20\) | \(-2\) | \(4\) |
\(260\) | \(-10\) | \(-1\) | \(1\) |
\(270\) | \(0\) | \(0\) | \(0\) |
\(300\) | \(30\) | \(3\) | \(9\) |
\(310\) | \(40\) | \(4\) | \(16\) |
\(330\) | \(60\) | \(6\) | \(36\) |
\(\sum d_{i} = 10\) | \(\sum d_{i}^{2} = 66\) |
The formula to calculate the standard deviation by step deviation method is given by:
\(\sigma = c \times \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\) where \(d_{i} = \frac{x_{i} - A}{c}\).
Substitute the known values in the above formula.
\(\sigma = 10 \times \sqrt{\frac{66}{6}- \left(\frac{10}{6}\right)^2}\)
\(=\) \(10 \times \sqrt{11 - (1.667)^2}\)
\(=\) \(10 \times \sqrt{11 - 2.779}\)
\(=\) \(10 \times \sqrt{8.221}\)
\(=\) \(10 \times 2.8672\)
\(=\) \(28.672\)
\(\approx\) \(28.67\)
Therefore, the standard deviation of the given data is \(28.67\).
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