The harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of individual observations.

Symbolically,

The calculation of harmonic mean becomes very tedious when a distribution has a large number of observations. In the case of grouped data, the harmonic mean is calculated by using the following formula:

Where n is the total number of observations.

Here, each reciprocal of the original figure is weighted by the corresponding frequency (f).

The main advantage of the harmonic mean is that it is based on all observations in a distribution and is amenable to further algebraic treatment. When we desire to give greater weight to smaller observations and less weight to the larger observations, then the use of harmonic mean will be more suitable.

As against these advantages, there are certain limitations of the harmonic mean. First, it is difficult to understand as well as difficult to compute. Second, it cannot be calculated if any of the observations is zero or negative. Third, it is only a summary figure, which may not be an actual observation in the distribution.

It is worth noting that the harmonic mean is always lower than the geometric mean, which is lower than the arithmetic mean. This is because the harmonic mean assigns lesser importance to higher values. Since the harmonic mean is based on reciprocals, it becomes clear that as reciprocals of higher values are lower than those of lower values, it is a lower average than the arithmetic mean as well as the geometric mean.

Example 1: Suppose we have three observations 4, 8 and 16. We are required to calculate the harmonic mean. Reciprocals of 4,8 and 16 are:respectively.

Example 2.18: Consider the following series:

 Class-interval 2-4 4-6 6-8 8-10 Frequency 20 40 30 10 Solution:

Let us set up the table as follows:

 Class-interval Mid-value Frequency Reciprocal of MV f x 1/x 2-4 3 20 0.3333 6.6660 4-6 5 40 0.2000 8.0000 6-8 7 30 0.1429 4.2870 8-10 9 10 0.1111 1.1111 Total 20.0641

Example 2: In a small company, two typists are employed. Typist A types one page in ten minutes while typist B takes twenty minutes for the same.

1. Both are asked to type 10 pages. What is the average time taken for typing one page?
2. Both are asked to type for one hour. What is the average time taken by them for typing one page?

Solution: Here Q-(i) is on arithmetic mean while Q-(ii) is on harmonic mean.

Example 3: It takes ship A 10 days to cross the Pacific Ocean; ship B takes 15 days and ship C takes 20 days.

1. What is the average number of days taken by a ship to cross the Pacific Ocean?
2. What is the average number of days taken by a cargo to cross the Pacific Ocean when the ships are hired for 60 days?

Solution: Here again Q-(i) pertains to simple arithmetic mean while Q-(ii) is concerned with the harmonic mean.

=          13.8 days approx.