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Harmonic Mean

For a given set of non-zero observations, harmonic mean is defined as the reciprocal of the A.M. of reciprocals of observations.

 

If a variable x assumes the values x1, x2, x3, ... xn, then the H.M. of x for a set of discrete data is given by

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Example
Find the harmonic mean for the numbers 2, 3, 5.
Solution
x1 = 2, x2 = 3, x3 = 5 and n = 3
Description: 97018.png 
Description: 97012.png 
When the given data is a grouped frequency distribution, the H.M. can be found using the relation
Description: 97006.png 
 
Example
Find the H.M. for the following data.
 
x 2 4 8 16
f 2 3 3 2
Solution
N = 2 + 3 + 3 + 2 = 10
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Description: 96984.png 
If there are two groups with n1 and n2 observations, and H1 and H2 are the respective H.M.s, then the combined H.M. is given by
Description: 96978.png 

 

Note: For any set of positive observations: A.M. ≥ G.M. ≥ H.M.
For any two positive observations: A.M. × H.M. = G.M.2

Weighted Average

When the observations under consideration have a hierarchical order of importance, we compute the weighted average, which could be either weighted A.M. or weighted G.M. or weighted H.M.

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Example
Find the weighted A.M. of the marks of a student given below.
Subject Marks Weight
English 65 1
Language 70 1
Physics 63 3
Chemistry 69 2
Mathematics 60 3
Statistics 71 2
Total 398 12
Solution
Subject Weight (w) Marks (x) Wx
English 1 65 65
Language 1 70 70
Physics 3 63 189
Chemistry 2 69 138
Mathematics 3 60 180
Statistics 2 71 142
Total 12 398 784
Description: 96873.png 

Merits

  1. It is easy to calculate.
  2. It is rigidly defined.
  3. It gives largest weight to the smallest items and it can be used whenever so desired.
  4. It is a useful average when we deal with average of rates.

Demerits

  1. It cannot be located by inspection.





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