Statistics - Variance

 
   
     
       

Variance

       

Understanding Variance in Statistics

       

          Variance measures how much the values in a data set deviate from the mean. It is a key concept in probability and statistics that quantifies data dispersion.        

     
     
       
         

Formula:

         

            \[             \sigma^2 = s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{\sum x_i^2 - \left(\sum x_i\right)^2/n}{n - 1}             \]          

         

Where:

         
               
  • \( x_i \): Each individual data value
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  • \( \bar{x} \): Mean of the data set
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  • \( n \): Number of observations
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  • \( s^2 \): Sample variance
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  • \( \sigma^2 \): Population variance
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Key Properties:

         
               
  • Always non-negative: \( \sigma^2 \geq 0 \)
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  • Zero variance means all values are equal
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  • Units are squared (e.g., if data is in cm, variance is in cm²)
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  • Sensitive to outliers due to squaring of deviations
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Applications:

         
               
  • Used in finance to measure investment risk
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  • Helps in identifying consistency in manufacturing
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  • Central to hypothesis testing and confidence intervals
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  • Foundation for standard deviation and many statistical models
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