At many times it may be necessary to calculate the weighted harmonic mean. For e.g. if we are not given only the speed of the aeroplane but the distances traveled also simple harmonic mean cannot be used.The weighted harmonic mean is calculated with the help of the following formula.

H.M. w [Σw/(1/a x w₁) (1/b x w₂) + (1/c x w₃)] or [Σw/Σ(w/x)].

Illustration :

A cyclist covers his first 5  km; at an average speed of 10 km/h. another km at 8km/h and the last 2  km. at 5 km find the average speed of the entire joule and verify your answer. 

Solution:

The Weighted harmonic mean would be appropriate here

H.M .w = [Σw/(1/a x w₁) + (1/b x w₂) + (1/c x w₃)]

The different speeds are 10km p/h, 8 km p/h, and 5 km p/h. And Hence a, b and c respectively are 108 and 5. The distances covered are respectively 53 and 2 km hence w₁, w₂ and w₃ 3, 53 and 2

H.M .w = 10/(1/10 x 5) + (1/8 X₃) + (1/2 X₂) = 10/1/2 + 3/8 + 2/5 = 10/ 51 / 40 10 x 40/ 51 = 7.84 km. p.h.

Thus the average speed for the entire journey is 7.84 km/h.

Verification :

• In 76.5 min. he covers 10 km.
• In 1 min. he would cover 10/76.5 x 60 = 7.84 km.
• In 60 min. he would cover 10/76.5 x 60 = 7.84 km.

Merits:

1)Its value is totally depend on every item of the series.

2)It tends itself to the algebraic manipulation.

3)The  problems relating to time &  rates it gives better results than other averages.

Limitations:

1)It is not easily understandable.

2)It is difficult to compute.

3)It gives the largest weight to smallest items this is generally not a desirable feature and as such this average is not very useful for the analysis of economic data.

4)Its value cannot be easily computed when there are both positive and negative items in a series or when one or more items are zero.

5)Because of these limitations the harmonic mean has little practical application and is not a good representation of a statistical  series unless the phenomenon is such where small items require to be given a very high weight age.