# HARMONIC MEAN

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 AM of the reciprocals of the observation.

Let the  variable "x" assume "n" values  as given below

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

For a grouped frequency distribution, we have

## 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, HM 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.

## Harmonic mean - Practice problems

Problem 1 :

Find the HM for 4, 6 and 10.

Solution :

For the given data, the formula to find HM is given by

Here n  =  3 and x  =  4, 6, 10

HM  =  3 / ( 1/4 + 1/6 + 1/10)

HM  =  3 / (0.25 + 0.17 + 0.10)

HM  =  5.77

Problem 2 :

Find the HM of the first three multiples of 5.

Solution :

The first three multiples of 5 are

5, 10 and 15

For the given data, the formula to find HM is given by

Here n  =  3 and x  =  5, 10, 15

HM  =  3 / ( 1/5 + 1/10 + 1/15)

HM  =  3 / (0.2 + 0.1 + 0.07)

HM  =  3 / 0.37

HM  =  8.11

Problem 3 :

Find the HM for the following distribution:

Solution :

For the given data, formula to find HM is given by

Here N  =  ∑f  =  10  and x  =  2, 4, 8, 6

HM  =  10 / ( 2/2 + 3/4 + 3/8 + 2/16)

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

HM  =  10 / 2.25

HM  =  4.44

After having gone through the stuff given above, we hope that the students would have understood "Harmonic-mean".

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