Statistics - Standard Deviation of Discrete Data Series

Quiz

When data is given alongwith their frequencies. Following is an example of discrete series:

Items	5	10	20	30	40	50	60	70
Frequency	2	5	1	3	12	0	5	7

For discrete series, the Standard Deviation can be calculated using the following formula.

Formula

$\sigma = \sqrt{\frac{\sum_{i=1}^n{f_i(x_i-\bar x)^2}}{N}}$

Where −

Problem Statement:

Calculate Standard Deviation for the following discrete data:

Items	5	15	25	35
Frequency	2	1	1	3

Solution:

Based on the given data, we have:

${ \bar x = \frac{5 \times 2 + 15 \times 1 + 25 \times 1 + 35 \times 3}{7} \\[7pt] = \frac {10 + 15 + 25 + 105}{7} = 22.15 }$

Items x	Frequency f	${\bar x}$	${x-\bar x}$	$f({x-\bar x})^2$
5	2	22.15	-17.15	580.25
15	1	22.15	-7.15	51.12
25	1	22.15	2.85	8.12
35	3	22.15	12.85	495.36
	${N=7}$			${\sum{f(x-\bar x)^2} = 1134.85}$

Based on the above mentioned formula, Standard Deviation $ \sigma $ will be:

${ \sigma =\sqrt{\frac{\sum_{i=1}^n{f_i(x_i-\bar x)^2}}{N}} \\[7pt] \, = \sqrt{\frac{1134.85}{7}} \, = 12.73}$

The Standard Deviation of the given numbers is 12.73.

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