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Standard  Deviation

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☼ Standard  Deviation  ( S. D.  or  \sigma  ) :

 (i) Ungrouped  Data : –

\sigma = \sqrt{\frac{\sum \left (x_{i}-\overline{x} \right ) ^{2}}{n}} =  \sqrt{\frac{\sum x_{i}^{2}}{n}- \left (\frac{\sum x_{i}}{n} \right )^{2}} = \sqrt{\frac{\sum d_{i}^{2}}{n}- \left (\frac{\sum d_{i}}{n} \right )^{2}}

 (ii) Discrete Data ( Grouped Data ) : –

   \sigma = \sqrt{\frac{\sum f_{i} \left (x_{i}-\overline{x} \right ) ^{2}}{n}} =  \sqrt{\frac{\sum f_{i} x_{i}^{2}}{n}- \left (\frac{\sum f_{i} x_{i}}{n} \right )^{2}} = \sqrt{\frac{\sum f_{i} d_{i}^{2}}{n}- \left (\frac{\sum f_{i} d_{i}}{n} \right )^{2}}

(iii)   \large \sigma  =  \sqrt{Variance}    or     Variance  = \large \sigma ^{2}

 

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