Statistics - Standard Deviation of Individual Data Series



When data is given on individual basis. Following is an example of individual series:

Items 5 10 20 30 40 50 60 70

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

Formula

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

Where −

  • ${x}$ = individual observation of variable.

  • ${\bar x}$ = Mean of all observations of the variable

  • ${N}$ = Number of observations

Example

Problem Statement:

Calculate Standard Deviation for the following individual data:

Items 14 36 45 70 105

Solution:

${X}$ ${\bar x}$ ${x- \bar x}$ ${(x - \bar x)^2}$
14 54 -40 1600
36 54 -18 324
45 54 -9 81
70 54 16 256
105 54 51 2601
${N=5}$     ${\sum{(x - \bar x)^2} = 4862}$

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

$ {\sigma = \sqrt{\frac{\sum{(x - \bar x)^2}}{N-1}} \\[7pt] \, = \sqrt{\frac{4862}{4}} \\[7pt] \, = \sqrt{\frac{4862}{4}} \\[7pt] \, = 34.86}$

The Standard Deviation of the given numbers is 34.86.

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