Statistics - Mean

 
   
     
       

Mean

       

Understanding the Arithmetic Mean in Statistics

       

          The Arithmetic Mean, often simply called the "mean" or "average," is the sum of all values in a dataset divided by the number of values. It is a commonly used measure of central tendency in statistics.        

        Mean Graph      
     
       
         

Formula:

         
           

              \[               \bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum x_i}{n}               \]            

         
         

Where:

         
               
  • \(x_1, x_2, \ldots, x_n\): the individual values in the dataset
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  • \(n\): the number of values in the dataset
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  • \(\bar{x}\): the arithmetic mean
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Key Properties of Arithmetic Mean:

         
               
  • Affected by every value in the dataset (including outliers)
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  • Sum of deviations from the mean is always zero: \( \sum (x_i - \bar{x}) = 0 \)
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  • Minimizes the sum of squared deviations: least squares property
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  • Applicable to both grouped and ungrouped data
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Applications of Mean:

         
               
  • Calculating average scores, salaries, or prices
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  • Analyzing trends in economics and finance
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  • Data normalization in machine learning
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  • Used in health, science, sports, and education analysis
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