Geometric mean | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-05

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Geometric_mean


00:03:59 1 Calculation
00:10:00 1.1 Relationship with logarithms
00:14:17 1.2 Relationship with arithmetic mean and mean-preserving spread
00:14:42 1.3 Computation in constant time
00:16:52 2 Properties
00:19:48 3 Applications
00:19:57 3.1 Proportional growth
00:22:00 3.2 Applications in the social sciences
00:23:20 3.3 Aspect ratios
00:26:48 3.4 Anti-reflective coatings
00:27:25 3.5 Spectral flatness
00:27:44 3.6 Geometry
00:28:52 3.7 Financial
00:29:26 3.8 Image Processing
00:29:40 4 See also
00:29:49 5 Notes and references
00:29:58 6 External links



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SUMMARY
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In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the nth root of the product of n numbers, i.e., for a set of numbers x1, x2, ..., xn, the geometric mean is defined as

  
    
      
        
          
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    {\displaystyle \left(\prod _{i=1}^{n}x_{i}\right)^{\frac {1}{n}}={\sqrt[{n}]{x_{1}x_{2}\cdots x_{n}}}}
  For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is, 
  
    
      
        
          
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    {\displaystyle {\sqrt {2\cdot 8}}=4}
  . As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the cube root of their product (1/8), which is 1/2, that is, 
  
    
      
        
          
            
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    {\displaystyle {\sqrt[{3}]{4\cdot 1\cdot 1/32}}=1/2}
  .
A geometric mean is often used when comparing different items—finding a single "figure of merit" for these items—when each item has multiple properties that have different numeric ranges. For example, the geometric mean can give a meaningful value to compare two companies which are each rated at 0 to 5 for their environmental sustainability, and are rated at 0 to 100 for their financial viability. If an arithmetic mean were used instead of a geometric mean, the financial viability would have greater weight because its numeric range is larger. That is, a small percentage change in the financial rating (e.g. going from 80 to 90) makes a much larger difference in the arithmetic mean than a large percentage change in environmental sustainability (e.g. going from 2 to 5). The use of a geometric mean normalizes the differently-ranged values, meaning a given percentage change in any of the properties has th ...