# Statistics - Standard Deviation of Discrete Data Series

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

 Items Frequency 5 10 20 30 40 50 60 70 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 −

• ${N}$ = Number of observations = ${\sum f}$.

• ${f_i}$ = Different values of frequency f.

• ${x_i}$ = Different values of variable x.

### Example

Problem Statement:

Calculate Standard Deviation for the following discrete data:

 Items Frequency 5 15 25 35 2 1 1 3

Solution:

Based on the given data, we have:

## Mean

${ \bar x = \frac{5 \times 2 + 15 \times 1 + 25 \times 1 + 35 \times 3}{7} \\[7pt] = \frac {10 + 15 + 25 + 105}{7} = 22.15 }$
Items
x
Frequency
f
${\bar x}$${x-\bar x}$$f({x-\bar x})^2$
5222.15-17.15580.25
15122.15-7.1551.12
25122.152.858.12
35322.1512.85495.36
${N=7}$  ${\sum{f(x-\bar x)^2} = 1134.85}$

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.