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12. Assumed mean method to find standard deviation of grouped data - Example

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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