Statistics - Standard Deviation of Discrete Data Series
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
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 | 5 | 15 | 25 | 35 |
---|---|---|---|---|
Frequency | 2 | 1 | 1 | 3 |
Solution:
Based on the given data, we have:
Mean
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:
The Standard Deviation of the given numbers is 12.73.