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13. Step deviation method to find standard deviation of grouped data - Example

Let us look at an example to find standard deviation of a grouped data by step deviation method.

Example:

The number of buckets required to fill given volume of water are given below:

Volume (in litres)	$2 - 4$	$4 - 6$	$6 - 8$	$8 - 10$	$10 - 12$
Number of buckets	$3$	$7$	$11$	$15$	$18$

Find its standard deviation using step deviation method.

Explanation:

Let the assumed mean be

$A = 7$ and the class width

$c = 2$ .

Volume	Number of buckets ( $f_{i}$ )	Midpoint ( $x_{i}$ )	Deviation $d_{i} = \frac{x_{i} - A}{c}$	$d_{i}^{2}$	$f_{i}d_{i}$	$f_{i}d_{i}^{2}$
$2 - 4$	$3$	$3$	$-2$	$4$	$-6$	$12$
$4 - 6$	$7$	$5$	$-1$	$1$	$-7$	$7$
$6 - 8$	$11$	$7$	$0$	$0$	$0$	$0$
$8 - 10$	$15$	$9$	$1$	$1$	$15$	$15$
$10 - 12$	$18$	$11$	$2$	$4$	$36$	$72$
	$\sum_{i = 1}^{5} f_{i} = 54$				$\sum_{i = 1}^{5} f_{i}d_{i} = 38$	$\sum_{i = 1}^{5} f_{i}d_{i}^{2} = 106$

The formula to calculate the standard deviation by step deviation method is given by:

$\sigma = c \times \sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}- \left(\frac{\sum f_{i} d_{i}}{N}\right) ^2}$ where

$N = \sum_{i = 1}^{n} f_{i}$ and

$d_{i} = \frac{x_{i} - A}{c}$ .

Substituting the known values in the above formula, we have:

$\sigma = 2 \times \sqrt{\frac{106}{54}- \left(\frac{38}{54}\right) ^2}$

$= 2 \times \sqrt{1.96 - \left(0.7 \right) ^2}$

$= 2 \times \sqrt{1.96 - 0.49}$

$= 2 \times \sqrt{1.47}$

$= 2 \times 1.212$

$=$

$2.424$

$\approx$

$2.42$

Therefore, the standard deviation of the given data is

$2.42$ .

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