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11. 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 mean method.

Example:

Case 1: Discrete data

Find the standard deviation of the following data by mean method.

$x$	$1$	$2$	$3$	$4$
$f$	$4$	$8$	$12$	$16$

Explanation:

We know that the formula for finding the arithmetic mean using direct method is given by:

$\overline x = \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{\sum_{i = 1}^{n} f_{i}}$

Where

$x_{i}$ is the midpoint of the class interval and

$f_{i}$ is the frequency.

Let us form a frequency distribution table.

$x_{i}$	$f_{i}$	$f_{i} x_{i}$
$1$	$4$	$4$
$2$	$8$	$16$
$3$	$12$	$36$
$4$	$16$	$64$
	$\sum_{i = 1}^{4} f_{i} = 40$	$\sum_{i = 1}^{4} f_{i} x_{i} = 120$

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

Mean

$\overline x = \frac{120}{40}$

$=$

$3$

Thus, the meanof the given data is

Now, let us find the standard deviation for the given data.

$x_{i}$	$f_{i}$	$d_{i} = x_{i} - \overline x$ $=$ $x_{i} - 3$	$d_{i}^{2}$	$f_{i}d_{i}^{2}$
$1$	$4$	$-2$	$4$	$16$
$2$	$8$	$-1$	$1$	$8$
$3$	$12$	$0$	$0$	$0$
$4$	$16$	$1$	$1$	$16$
	$\sum_{i = 1}^{4} f_{i} = 40$			$\sum_{i = 1}^{4} f_{i} d_{i}^{2} = 40$

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

$\sigma$

$=$

$\sqrt{\frac{\sum f_{i} d_{i}^{2}}{N}}$ where

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

Substitute the required values in the above formula.

$\sigma$

$=$

$\sqrt{\frac{40}{40}}$

$=$

$\sqrt{1}$

$=$

$1$

Therefore, the standard deviation of the given data is

$1$ .

Case 2: Continuous data