Weighted Harmonic Mean

The harmonic mean is the reciprocal of the arithmetic mean of a set of data values. The weighted harmonic mean is a sub-unit of harmonic mean in which the sum of all weights are equal to one (or say 100%). 

This material is about the weighted harmonic mean. You will find brief notes on the concept of weighted harmonic mean, a thorough explanation of its properties and importance in calculating certain events, the formula for weighted harmonic mean, and so on.  

Let’s start by defining the weighted harmonic mean.

Definition of weighted harmonic mean

For a given equation where, frequencies ‘f’ represents the weights ‘w’, the harmonic mean would be computed as:

If x1, x2,………….., xn are n items with corresponding frequencies f1, f2,………, fn, then the weighted harmonic mean is:

HM = N / [(f1/x1) + (f2/x2) + (f3/x3) + (f4/x4)….+ (fn/xn)

Where N = total number of observations

f = frequency of the observations

and x = observations

Formula for weighted harmonic mean

HMw = ∑w / ∑(w/a)

HM = w1 + w2 + w3+ …………….+wn) / [(w1/a1)+(w2/a2)+(w3/a3)+…+(wn/an)]

Where w = weights assigned to frequencies and a = observations

Example:

An individual buys three pens of different qualities. With the relevant data given below, calculate the average price per pencil.

Solution:

Now, assume that we do not know about the concept of harmonic mean. Then, we can solve this problem as

Average price per pen = Total money spent / Total number of pens sold

 = (50+30+20) / (50+20+10)

= 1.25

Now, by applying weighted harmonic mean, let’s see the result.

Note that the different amount spent is weights here.

Using

HMw = ∑w / ∑(w/x)

= (50+30+20) / (50/1+30/1.5+20/2)

= 1.25

We can see that results from both methods are the same. Hence, we can use the concept of harmonic mean to our daily problems.

Properties of Harmonic Mean

1. If all observations are the same, then the harmonic mean equals a single observation. For example, if we have been given a series where there are five observations and all the observations are the same, such as 2, 2, 2, 2, 2, then the harmonic mean for the given series will be 2. 

1. Change in the harmonic mean is not equal to change in origin. For example,

X = 2, 3, 4, 2

HM = 2.526

Let us change the origin as = X+1

X+1 = 3, 4, 5, 3

HM = 3.582

Change in HM due to change in origin = 3.582 – 2.526 = 1.056

This proves that change in HM is not equal to change in origin.

1. The harmonic mean can change when the scale changes.

For example,

X = 2, 3, 4, 2

HM = 2.526

Let us change the scale here as = X*3

X*3 = 6, 9, 12, 6

HM = 7.578

Change in HM due to change in scale = 2.526*3 = 7.578

This change in harmonic mean is equal to a change in scale. Therefore we can conclude that change in scale is equal to change in harmonic mean. 

1. Combined harmonic mean

= (n1+n2 + n3 + …………nn) / [(1/HM1)+(1/HM2)+(1/HM3)+…+(1/HMn)]

Advantages of harmonic mean

• It is based on all the observations in a data set
• It is an appropriate type for calculating the average rates and ratios
• It is not affected much by sampling variability
• A mathematical formula has rigorously defined it 

Disadvantages of harmonic mean

• It cannot be calculated if the value of any one of the observations is zero
• This method gives too much weight to smaller observations
• This method is difficult to understand, and the calculation procedure is also difficult as compared to other means

Relationship among arithmetic mean, geometric mean and harmonic mean

The following are the formulae for the three classifications of means:

Arithmetic Mean = (a1 + a2 + a3 +…..+an ) / n

Harmonic Mean = n / [(1/a1)+(1/a2)+(1/a3)+…+(1/an)]

Geometric Mean = √ab

If G is the geometric mean, H is the harmonic mean, and A is the arithmetic mean, their relationship is given by,

AM >= GM >= HM

[(a+b)/2 >= √ab >= 2ab / (a+b)]

Here a and b are positive numbers

Conclusion

In this article, we studied the properties of the weighted harmonic mean in detail. We covered several related topics, such as the functions that can be performed with the help of the weighted harmonic mean and weighted harmonic mean formula, along with the solved examples.