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Let 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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