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# Harmonic Mean (H.M.)

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.

## (a) H.M. for Ungrouped data

Let  x1, x2, ..., 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

## (b)H.M. for Discrete Grouped data:

For a frequency distribution Example 5.12

The following data is obtained from the survey. Compute H.M Solution: ## (c)H.M. for Continuous data: Where xi is the mid-point of the class interval

Example 5.13

Find the harmonic mean of the following distribution of data Solution: ## Merits of H.M:

·           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

## Limitations of H.M:

·           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

### When to use?

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.

### Relationship among the averages:

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

### Various measures of central tendency Tags : Formula, Solved Example Problems , 11th Statistics : Chapter 5 : Measures of Central Tendency
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11th Statistics : Chapter 5 : Measures of Central Tendency : Harmonic Mean (H.M.) | Formula, Solved Example Problems