Harmonic Mean

In Statistics, to represent the data or value in series, the measure of central tendency is used. A measure of central tendency is a single value that describes the way that the group of data clusters around a central value. It defines the centre of the data set. There are three measures of central tendency. They are mean, median, and mode. In this article, you will learn one of the important types of mean called “Harmonic Mean” with the definition, formula, and examples in detail.

Table of Contents:

Harmonic Mean Definition

The Harmonic Mean (HM) is defined as the reciprocal of the average of the reciprocals of the data values.. It is based on all the observations, and it is rigidly defined. Harmonic mean gives less weightage to the large values and large weightage to the small values to balance the values correctly. In general, the harmonic mean is used when there is a necessity to give greater weight to the smaller items. It is applied in the case of times and average rates.

Harmonic Mean Formula

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)]

How to Find a Harmonic Mean?

If a, b, c, d, … are the given data values, then the steps to find the harmonic mean are as follows:

Step 1: Calculate the reciprocal of each value (1/a, 1/b, 1/c, 1/d, …)

Step 2: Find the average of reciprocals obtained from step 1.

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

Relationship Between Arithmetic Mean, Geometric Mean and Harmonic Mean

The three means such as arithmetic mean, geometric mean, harmonic means are known as Pythagorean means. The formulas for three different types of means are:

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

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

Geometric Mean = \(\sqrt[n]{a_{1}.a_{2}.a_{3}…a_{n}}\)

If G is the geometric mean, H is the harmonic mean, and A is the arithmetic mean, then the relationship between them is given by:

\(G = \sqrt{AH}\)

Or

G2 = A.H

Weighted Harmonic Mean

Calculating weighted harmonic mean is similar to the simple harmonic mean. It is a special case of harmonic mean where all the weights are equal to 1. 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

\(HM_{w}=\frac{\sum_{i=1}^{n}w_{i}}{\sum_{i=1}^{n}\frac{w_{i}}{x_{i}}}\)

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

HMw = N / [ (f1/x1) + (f2/x2) + (f3/x3)+ ….(fn/xn) ]

Note:

  1. f values are considered as weights
  2. For continuous series, mid-value = (Lower limit + Upper limit)/2 and is taken as x.

Harmonic Mean Uses

The main uses of harmonic means are as follows:

  • The harmonic mean is applied in the finance to the average multiples like price-earnings ratio
  • It is also used by the market technicians in order to determine the patterns like Fibonacci Sequences

Merits and Demerits of Harmonic Mean

The following are the merits of the harmonic mean:

  • It is rigidly confined.
  • It is based on all the views of a series, i.e. it cannot be computed by ignoring any item of a series.
  • It is able to advance the algebraic method.
  • It provides a more reliable result when the results to be achieved are the same for the various means adopted.
  • It provides the highest weight to the smallest item of a series.
  • It can also be measured when a series holds any negative value.
  • It produces a skewed distribution of a normal one.
  • It produces a curve straighter than that of the A.M and G.M.

The demerits of the harmonic series are as follows:

  • The harmonic mean is greatly affected by the values of the extreme items
  • It cannot be able to calculate if any of the items is zero
  • The calculation of the harmonic mean is cumbersome, as it involves the calculation using the reciprocals of the number.

Harmonic Mean Examples

Example 1:

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 2:

Calculate the harmonic mean for the following data:

x 1 3 5 7 9 11
f 2 4 6 8 10 12

Solution:

The calculation for the harmonic mean is shown in the below table:

x f 1/x f/x
1 2 1 2
3 4 0.333 1.332
5 6 0.2 1.2
7 8 0.143 1.144
9 10 0.1111 1.111
11 12 0.091 1.092
N =42 Σ f/x = 7.879

The formula for weighted harmonic mean is

HMw = N / [ (f1/x1) + (f2/x2) + (f3/x3)+ ….(fn/xn) ]

HMw = 42 / 7.879

HMw = 5.331

Therefore, the harmonic mean, HMw is 5.331.

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Frequently Asked Questions on Harmonic Mean

Define harmonic mean.

The harmonic mean is defined as the reciprocal of the average of the reciprocals of the given data values.

Mention the steps to calculate the harmonic mean.

The steps to calculate the harmonic mean are as follows:
Step 1: Find the reciprocal of the given values
Step 2: Calculate the average for the reciprocals obtained in step 1.
Step 3: Finally, calculate the reciprocal of the average obtained from step 2.

What is the relationship between AM, GM, and HM?

If AM, GM, and HM are the arithmetic mean, geometric mean and harmonic mean, respectively, then the relationship between AM, GM and HM is GM2 = AM × HM

What is the harmonic mean of a and b?

The harmonic mean of a and b is 2ab/(a+b).
As “a” and “b” are the two data values, then the harmonic mean is written as
H.M = 2 /[(1/a)+(1/b)]
H.M = 2/[(a+b)/ab]
H.M = 2ab/(a+b)

Calculate the harmonic mean of 2 and 4.

Harmonic Mean = [2(2)(4)]/(2+4)
Harmonic Mean= 16/6
Harmonic Mean = 2.67
Hence, the harmonic mean of 2 and 4 is 2.67.

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