Statistics - Geometric Mean

 
   
     
       

Geometric Mean

       

Understanding the Geometric Mean in Statistics

       

          The Geometric Mean (G.M.) is the average of a set of positive numbers calculated by multiplying them together and then taking the \(n^{\text{th}}\) root (where \(n\) is the total number of values). It is especially useful for data involving rates, ratios, or percentages.        

     
     
       
         

Formula:

         
           

              \[               G.M. = \sqrt[n]{x_1 \cdot x_2 \cdot \cdots \cdot x_n}               \]            

         
         

Where:

         
               
  • \(x_1, x_2, ..., x_n\): positive values in the data set
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  • \(n\): total number of values
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Key Properties of Geometric Mean:

         
               
  • Only defined for positive numbers
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  • Less affected by extreme values compared to Arithmetic Mean
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  • \(G.M. \leq A.M.\), with equality only if all values are the same
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  • Used when values are multiplicative or in exponential growth
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Applications of Geometric Mean:

         
               
  • Calculating average growth rates in finance (e.g., interest, returns)
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  • Finding average ratios or percentages (e.g., inflation, population)
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  • Useful in geometry and physics for scaling problems
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  • Applied in environmental studies (e.g., average pollutant levels)
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