HARMONIC MEAN

Harmonic mean is one of the measures of central tendency which can be defined as follows.

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

Let the  variable x assume n values  as given below.

x₁, x₂, x₃, ............x_n

All the above values are being non zero values, then the harmonic mean of x is given by

H = n/∑(1/x_i)

For a grouped frequency distribution, we have

H = N/∑(f_i/x_i)

Properties of Harmonic Mean

1. If all the observations taken by a variable are constants, say k, then the HM of the observations is also k.

2. If there are two groups with n₁ and n₂ observations and H₁ and H₂ as respective HM’s than the combined HM is given by

3. Like arithmetic mean, harmonic mean also possess some mathematical properties.

4. It is rigidly defined.

5. It is based on all the observations.

6. It is difficult to comprehend.

7. It is difficult to compute.

8. It has limited applications for the computation of average rates and ratios and such like things. 

Practice Problems

Problem 1 :

Find the harmonic mean for 4, 6 and 10.

Solution :

Formula to find harmonic mean :

H = n/∑(1/x_i)

Fitting the given data in to the above formula, we get

H = 3/(1/4 + 1/6 + 1/10)

= 3/(0.25 + 0.17 + 0.10)

= 5.77

Problem 2 : 

Find the harmonic mean of the first three multiples of 5.

Solution :

The first three multiples of 5 are 5, 10 and 15.

Formula to find harmonic mean :

H = n/∑(1/x_i)

Fitting the given data in to the above formula, we get

H = 3/(1/5 + 1/10 + 1/15)

= 3/(0.25 + 0.17 + 0.10)

= 3/(0.2 + 0.1 + 0.07)

= 3/0.37

= 8.11

Problem 3 :

Find the HM for the following distribution:

x      :      2      4      8      16

f       :      2      3      3      2

Solution : 

Formula to find geometric mean for a grouped frequency distribution :

H = N/∑(f_i/x_i)

Fitting the given data in to the above formula, we get

= 10/(2/2 + 3/4 + 3/8 + 2/16)

= 10/(1 + 0.75 + 0.375 + 0.125)

= 10/2.25

= 4.44

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