UPSKILL MATH PLUS
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Learn moreLet us look at an example to find standard deviation of ungrouped data by assumed mean method.
Example:
Find the standard deviation of the data \(5\), \(8\), \(10\), \(11\) and \(12\) which represents the number cookies in \(5\) bottles by assumed mean method.
Explanation:
Let \(n\) represent the number of values in the data.
\(n\) \(=\) \(5\)
Let \(\overline x\) represent the mean of the given data.
Mean \(\overline x = \frac{\text{Sum of all the observations}}{\text{Total number of observations}}\)
\(\overline x\) \(=\) \(\frac{5 + 8 + 10 + 11 + 12}{5}\)
\(=\) \(\frac{46}{5}\)
\(=\) \(9.2\)
Here, the mean is not an integer value.
So, let us find the standard deviation by assumed mean method.
Let \(A\) be the assumed mean, which is the middle most value.
Here, \(A\) \(=\) \(10\)
Let \(x_{i}\) represent the marks scored by each student.
\(x_{i}\) | \(d_{i} = x_{i} - A\) \(=\) \(x_{i} - 10\) | \(d_{i}^{2}\) |
\(5\) | \(-5\) | \(25\) |
\(8\) | \(-2\) | \(4\) |
\(10\) | \(0\) | \(0\) |
\(11\) | \(1\) | \(1\) |
\(12\) | \(2\) | \(4\) |
\(\sum d_{i} = -4\) | \(\sum d_{i}^{2} = 34\) |
The formula to calculate the standard deviation by assumed mean method is given by:
\(\sigma = \sqrt{\frac{\sum d_{i}^{2}}{n}- \left(\frac{\sum d_{i}}{n}\right)^2}\) where \(d_{i} = x_{i} - A\).
Substitute the known values in the above formula.
\(\sigma = \sqrt{\frac{34}{9}- \left(\frac{-4}{9}\right)^2}\)
\(=\) \(\sqrt{3.778 - (-0.444)^2}\)
\(=\) \(\sqrt{3.778 - 0.198}\)
\(=\) \(\sqrt{3.58}\)
\(=\) \(1.892\)
\(\approx\) \(1.89\)
Therefore, the standard deviation of the given data is \(1.89\).
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