Harmonic Mean

We are familiar with calculating the arithmetic mean, in which the sum of values is divided by the number of values. Now in this article let us study what is harmonic mean in statistics, properties of the harmonic mean(HM), harmonic mean examples…A simple way to define harmonic mean is: It is the reciprocal of the arithmetic mean of the reciprocals of the observations. Harmonic mean is used to calculate the average of a group of numbers. The number of elements will be averaged and divided by the sum of the reciprocals of the elements.The most common examples of ratios are that of speed and time,work and time etc.

What is the Definition of the Term Harmonic Mean?

The Harmonic Mean (HM) is defined as the reciprocal of the arithmetic mean of the reciprocals of the observations. Harmonic mean gives less weightage to the larger values and more weightage to the smaller values to balance the values properly. The harmonic mean is generally used when there is a necessity to give greater weight to the smaller items. The harmonic mean is often used to calculate the average of the ratios or rates of the given values. It is the most appropriate measure for ratios and rates because it equalizes the weights of each data point.

The harmonic mean is a type of Pythagorean mean. When we divide the number of terms in a data series by the sum of all the reciprocal terms we get the harmonic mean. The value of the harmonic mean will always be the lowest as compared to the geometric and arithmetic mean.

Formula of Harmonic Mean

Since the harmonic mean is the reciprocal of the average of reciprocals, the formula to define the harmonic mean “HM” is given as follows:

If x1, x2, x3,…, xn are the individual items up to n terms, then,

Harmonic Mean, HM = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]

Thus, the formula to define the harmonic mean “H” is given as follows

Where,

n is a total number of terms

 x1, x2, x3,…, xn are the individual values up to nth terms.

Solved Examples

Example 1: Find the harmonic mean of the following data {8, 9, 6, 11, 10, 5}?

Solution:

Given data: {8, 9, 6, 11, 10, 5}

We have harmonic mean formula as:

Harmonic Mean(H) = n / [(1/x1)+(1/x2)+(1/x3)+…+(1/xn)]

n = 6

So Harmonic mean = 6/(⅛)+(1/9)+(⅙)+(1/11)+(1/10)+(⅕)

          = 6/0.7936

           = 7.560

Harmonic mean(H) = 7.560

Example 2: Find the harmonic mean for data 2, 5, 7, and 9.

Solution:

Given data: 2, 5, 7, 9

Step 1: Finding the reciprocal of the values:

½ = 0.5

⅕ = 0.2

1/7 = 0.14

1/9 = 0.11

Step 2: Calculate the average of the reciprocal values obtained from step 1.

Here, the total number of data values is 4.

Average = (0.5 + 0.2 + 0.143 + 0.11)/4

Average = 0.953/4

Step 3: Finally, take the reciprocal of the average value obtained from step 2.

Harmonic Mean = 1/ Average

Harmonic Mean = 4/0.953

Harmonic Mean = 4.19

Hence, the harmonic mean for the data 2, 5, 7, 9 is 4.19.

Example 3: What is the harmonic mean of 1, 2, and 4?

Solution: 

The reciprocals of 1, 2 and 4 are:

1/1 = 1,   

1/2 = 0.5,   

1/4 = 0.25

Now add them up:

1 + 0.5 + 0.25 = 1.75

Divide by how many:

Average = 1.75/3

The reciprocal of that average is our answer:

Harmonic Mean = 3/1.75

 = 1.714 (to 3 places)

Concept of Harmonic Mean

Harmonic mean is a type of numerical average that is usually used in situations when the average rate or rate of change needs to be calculated. Harmonic mean is one of the three Pythagorean means. The remaining two are the arithmetic mean and the geometric mean. These three means or averages are very important as they see widespread use in the field of geometry and music.

If we are given a data series or a set of observations then the harmonic mean can be defined as the reciprocal of the average of the reciprocal terms. This implies that the harmonic mean of a particular set of observations is the reciprocal of the arithmetic mean of the reciprocals.

Weighted Harmonic Mean

The weighted harmonic mean is a special case of harmonic mean where all the weights are equal to 1. It is similar to the simple harmonic mean. If the set of weights such as w1, w2, w3, …, wn connected with the sample space x1, x2, x3,…., xn, then the weighted harmonic mean is defined by

If the frequencies “f” is supposed to be the weights “w”, then the harmonic mean is calculated as follows:

If x1, x2, x3,…., xn are n items with corresponding frequencies f1, f2, f3, …., fn, then the weighted harmonic mean is

Also written as 

Where w = weight and x is the variable

Properties of Harmonic Mean

Harmonic means possess few properties which make them different from other types of means. 

1. For all the observations at constant, say c, then the harmonic means calculated of the observations will also be c.

2. The harmonic mean can also be evaluated for the series having any negative values.

3. If any of the values of a given series is 0 then its harmonic mean cannot be determined as the reciprocal of 0 doesn’t exist.

4. If in a given series all the values are neither equal nor any value is zero, then the harmonic mean calculated will be lesser than the geometric mean and arithmetic mean.

5. When compared to geometric mean and arithmetic mean, the harmonic mean possesses the least value i.e. AM > GM > HM.

Uses of Harmonic Mean

The harmonic mean is used in the following ways:

• In calculating average prices, average speed, etc under certain conditions.

• The harmonic mean is useful in the finance sector to calculate the average multiples like the price-earnings ratio

• It is also used in computing Fibonacci Sequences.

Merits and Demerits of Harmonic Mean

The merits and demerits of the harmonic mean are as follows:

1. Merits of Harmonic Mean

The harmonic mean is completely based on observations and is very useful in averaging certain types of rates. Other merits of the harmonic mean are given below.

As the value of harmonic mean remains fixed thus, it is rigidly defined.

Even if there is a sample fluctuation, the harmonic mean does not get significantly affected. Basically, it is least affected by fluctuation in sampling.

All items of the series are required to determine the harmonic mean. It is based on all observations.

It gives greater importance to small items.

It is capable of further algebraic treatment.

2. Demerits of Harmonic Mean

To calculate the harmonic mean, all elements of the series must be known. In case of unknown elements, we cannot determine the harmonic mean. Given below are other demerits of harmonic mean.

• The method to calculate the harmonic mean can be lengthy and complicated. It is difficult to calculate.

• If any term of the given series is 0 then the harmonic mean cannot be calculated.

• The extreme values in a series greatly affect the harmonic mean.

• It does not give equal weight to every item.

• It may not be represented in the actual data.

• It is not defined for a negative value.

Steps to Calculate Harmonic Mean

Step 1: Calculate the reciprocal (1/value) for every value.

Step 2: Find the average of those reciprocals, by just adding them and divide by the number of total values

Step 3: Then do the reciprocal of that average.

Solve Example 

1. What is the harmonic mean of 4, 5, anThe harmonic the harmonic arithmetic, and 10?

Solution:

The reciprocals of 4, 5 and 10 are:

1/4 = 0.25 ;  1/5 = 0.20 ;   1/10 = 0.10

Now add them

0.25 + 0.20 + 0.10 = 0.55

Total values are 3 so divide it by 3

Average = 0.55/3

The reciprocal of that average is our answer:

Harmonic Mean = 3/0.55

     = 5.454 (to 3 places)