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Geometric Mean

The Geometric Mean is defined as the nth root of the product of n observations. Thus, if a variable x assumes n values x1, x2, x3, ... xn, all the values being positive, then the G.M. of x for a set of discrete data is given by

Description: 94331.png 

What is the G.M. for the numbers 8, 24 and 40?
x1 = 8, x2 = 24, x3 = 40 and n = 3
Description: 94348.png 
When the given data is a grouped frequency distribution, the G.M. can be found using the relation
Description: 94373.png 


Find the G.M. for the following distribution:
X 2 4 8 6
f 2 3 3
x1 = 2, x2 = 4, x3 = 8, x4 = 16 and N = 2 + 3 + 3 + 2 = 10
Description: 94389.png 

Properties of Geometric Mean

  1. Let xi, where, i = 1, 2, 3, … n be a set of observations, then Description: 94463.png.
  2. If z = xy, then G.M. of z = (G.M. of x) × (G.M. of y)
  3. Description: 94469.png


  1. It is the only average that can be used to indicate the rate of change or ratios.
  2. It is very simple and lends itself to algebraic treatment.
  3. It is very useful in the construction of index numbers.
  4. It is not much affected by the fluctuations of sampling.
  5. It is based on all the observations.
  6. It gives less weight to large items and large weights to small items.


  1. It cannot be easily understood.
  2. It is relatively difficult to compute as it requires some special knowledge of logarithms.
  3. It cannot be calculated when any value is zero or negative.
  4. It may be a value which does not correspond to actual value.
  5. It cannot be obtained by inspection.

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