Statistics - Harmonic Mean

 
   
     
       

Harmonic Mean

       

Understanding the Harmonic Mean in Statistics

       

          The Harmonic Mean (H.M.) is a type of average used when values are defined in relation to some unit, like speed (distance/time), rates, or ratios. It is particularly useful when all data values contribute equally to a whole.        

     
     
       
         

Formula:

         
           

              \[               H.M. = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}} \quad \text{and} \quad \bar{x} \ (\text{A.M.}) \geq G.M. \geq H.M.               \]            

         
         

Where:

         
               
  • \(x_1, x_2, \ldots, x_n\): values in the dataset (non-zero positive numbers)
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  • \(n\): total number of values
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Key Properties of Harmonic Mean:

         
               
  • Only defined for positive, non-zero values
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  • Most appropriate for averaging rates, such as speed or efficiency
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  • \( H.M. \leq G.M. \leq A.M. \), with equality if all values are equal
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  • Sensitive to very small values in the dataset
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Applications of Harmonic Mean:

         
               
  • Average speed (e.g., round-trip travel with different speeds)
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  • Financial ratios such as Price/Earnings (P/E) averaging
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  • Used in meta-analysis and data science models
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  • Helps when combining rates (like work done per time unit)
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