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7. Direct method to find standard deviation of ungrouped data - Example

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.

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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