Harmonic Mean is defined as the reciprocal of the arithmetic mean of reciprocals of the observations.
(a) H.M. for Ungrouped data
(b) H.M. for Discrete Grouped data:
(c) H.M. for Continuous data:

Harmonic Mean (H.M.)

Harmonic Mean is defined as the reciprocal of the arithmetic mean of reciprocals of the observations.

Let *x*1, *x*2, ..., *xn* be the *n* observations then the harmonic mean is defined as

Example 5.11

A man travels from Jaipur to Agra by a car and takes 4 hours to cover the whole distance. In the first hour he travels at a speed of 50 km/hr, in the second hour his speed is 64 km/hr, in third hour his speed is 80 km/hr and in the fourth hour he travels at the speed of 55 km/hr. Find the average speed of the motorist.

*Solution:*

Average speed of the motorist is 60.5km/hr

For a frequency distribution

Example 5.12

The following data is obtained from the survey. Compute H.M

*Solution:*

Where *xi* is the mid-point of the class interval

Example 5.13

Find the harmonic mean of the following distribution of data

*Solution:*

· It is rigidly defined

· It is based on all the observations of the series

· It is suitable in case of series having wide dispersion

· It is suitable for further mathematical treatment

· It gives less weight to large items and more weight to small items

· It is difficult to calculate and is not understandable

· All the values must be available for computation

· It is not popular due to its complex calculation.

· It is usually a value which does not exist in series

*Harmonic mean is used to calculate the average value when the values are expressed as value/unit. Since the speed is expressed as km/hour, harmonic mean is used for the calculation of average speed.*

In any distribution when the original items are different the A.M., G.M. and H.M would also differ and will be in the following order:

A.M. ≥ G.M ≥ H.M

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