Statistics - Standard Deviation

 
   
     
       

Standard Deviation

       

Understanding Standard Deviation in Statistics

       

          Standard Deviation (SD) measures the amount of variation or dispersion in a set of numerical values. It tells us how far individual data points deviate from the mean of the data set.        

     
     
       
         

Formula:

         

            \[             s = \sqrt{\text{variance}} = \sqrt{s^2} = \sqrt{\frac{1}{n} \sum_{i=1}^{n}(x_i - \bar{x})^2}             \]          

         

Key Terms:

         
               
  • \( s \): standard deviation
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  • \( \bar{x} \): mean of the data
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  • \( x_i \): individual value in the dataset
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  • \( n \): total number of values
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Key Properties:

         
               
  • Always non-negative (\( s \geq 0 \))
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  • A low SD indicates that values are close to the mean
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  • A high SD shows greater spread or variability
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  • Used for both population and sample data
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Applications:

         
               
  • Risk analysis in finance (volatility of returns)
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  • Quality control in manufacturing
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  • Performance analysis in sports and education
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  • Understanding variation in scientific experiments
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