Harmonic Mean

The Harmonic Mean (HM) is a type of average used primarily when dealing with rates, ratios, or speeds. It is defined as the reciprocal of the arithmetic mean of the reciprocals of the given set of values. The harmonic mean gives equal weight to each data point and is especially useful when the data values are expressed as fractions or when we need to find an average rate of change.

Mathematically, if x₁, x₂, x₃, …, x_n​ are n non-zero positive observations, the harmonic mean is given by:

The harmonic mean is one of the three Pythagorean means, the other two being the Arithmetic Mean and the Geometric Mean. Among these, the harmonic mean is always the smallest. It is most appropriate for datasets involving quantities like speed, density, or efficiency, where values are defined in terms of a ratio or rate.

Harmonic Mean Formula

The harmonic mean of the data set is calculated using the formula. Let x₁, x₂, x₃, x₄, ... x_n is the n terms of the given data then the Harmonic Mean of the given data can be calculated by the formula,

Harmonic Mean (H.M) = \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}

Proof of Harmonic Mean Formula

As harmonic mean is the inverse of the arithmetic mean of reciprocal data terms.

So the arithmetic mean for the data x₁, x₂, x₃, ..., x_n is,

Arithmetic mean = \frac{x_1+x_2+x_3+...+x_n}{n}

In harmonic mean, we consider the reciprocal of data values.

So the arithmetic mean of reciprocal data 1/x₁, 1/x₂, 1/x₃, ..., 1/x_n  is,

Arithmetic mean for reciprocal data = \frac{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}{n}  . . .(1)

It is known that the Harmonic mean is the inverse of the arithmetic mean of reciprocal data values from eq (1)

Harmonic Mean = Inverse of the Arithmetic Mean of Reciprocal Data

⇒ Harmonic Mean = (\frac{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}{n})^{-1}

⇒ Harmonic Mean = \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+...+(1/x_n)}

This is the harmonic mean formula of the given data set.

Harmonic Mean of Two Numbers

We can find the harmonic mean of the two numbers by using the formula discussed above, suppose the two numbers are, a and b

n = 2

Reciprocal of a and  b is 1/a and 1/b

HM = 2/[1/a + 1/b]

HM = (2ab)/(a + b)

Weighted Harmonic Mean

It is similar to the Harmonic mean but in addition, to the normal value we take the weight value of the data set. If the weights of each data set is equal to 1 then it is the same as the Harmonic mean formula. Weighted Harmonic mean is calculated for the given set of weights of the data set, Suppose the weights of the data set are, w₁,w₂,w₃,w₄,...,w_n and their values are x₁, x₂, x₃, x₄, ..., x_n is, then the weighted harmonic mean formula,

Weighted Harmonic Mean = \frac{\sum_{i=1}^{i=n}W_i}{\sum_{i=1}^{i=n}\frac{w_i}{x_i}}

This formula is used when the weight of the given data set is given.

Harmonic Mean Example

We can understand the concept of the harmonic mean by studying the example discussed below.

For example, find the harmonic mean of the data set (2, 4, 8, 16). 

Solution:

Given data set, (2, 4, 8, 16)

Reciprocal of the data set, (1/2, 1/4, 1/8, 1/16)

Finding the sum of reciprocal value = (1/2 + 1/4 + 1/8 + 1/16)

⇒ Harmonic Mean = 4/(8/16 + 4/16 + 2/16 + 1/16)

⇒ Harmonic Mean = 4/(15/16)

⇒ Harmonic Mean = 64/15

Thus, the required harmonic mean is 64/15.

Relation Between AM, GM, and HM

The relation between AM, GM, and the HM of the data set is that the square of the geometric mean is equal to the product of the arithmetic mean and the harmonic mean.

Its proof is discussed below in the article.

For the data x₁, x₂

Arithmetic Mean(AM) = (x₁ + x₂)/2

Harmonic Mean(HM) = 2/((1/x₁) + (1/x₂))

Geometric Mean(GM) = ²√(x_{1 .} x₂)

Taking the square of GM equality,

(GM)² = x_{1 .} x₂

Now,

HM = 2 x_{1 .} x₂ (1/(x₁ + x₂))

⇒ HM = GM² (2/(x₁ + x₂))

⇒ HM = GM² (1/ AM)

HM × AM = GM²

How to Find Harmonic Mean?

The harmonic mean of the data set can easily be found using the steps discussed below,

Step 1: Find the total number of the data set given and mark it as, (n)

Step 2: Find the reciprocal of the given data set.

Step 3: Find the sum of all the reciprocal elements.

Step 4: Divide n by the sum of reciprocal values to get the required Harmonic Mean of the data set.

We can understand this concept with the help of the example discussed below,

Example: Find the harmonic mean of the data set (3, 6, 9). 

Solution:

Given data set, 3, 6, 9)

Step 1: Here, n = 9

Step 2: Reciprocal of the data set, (1/3, 1/6, 1/9)

Step 3: Finding the sum of reciprocal value = (1/3 + 1/6 + 1/9) = (6/18 + 3/18 + 2/18) = 11/18

Step 4: Finding Harmonic Mean i.e., H.M.= 3/(11/18)

⇒ H.M. = 54/11

Tnus, the harmonic mean of the data set is 54/11.

Harmonic Mean vs Geometric Mean

Harmonic mean and Geometric mean are the measure of the central tendencies they are both Pythagorean mean and the basic difference between them is discussed in the table below,

Harmonic Mean vs Arithmetic Mean

Harmonic mean and Arithmetic mean are the measure of the central tendencies they are both Pythagorean mean and the basic difference between them is discussed in the table below,

Solved Examples on Harmonic Mean

Example 1: Find the Harmonic Mean for the data 10, 20, 5, 15, 10.

Solution:

Given,

10, 20, 5, 15, 10

n = 5 

Harmonic Mean= \frac{n}{(1/x_1)+(1/x_2)+(1/x_3)+(1/x_4)+...+(1/x_n)}

⇒ Harmonic Mean = \frac{5}{(1/10)+(1/20)+(1/5)+(1/15)+(1/10)}

⇒ Harmonic Mean = 5/(0.1 + 0.05 + 0.2 + 0.06 + 0.1)
⇒ Harmonic Mean = 5/0.51
⇒ Harmonic Mean = 9.8

Hence, the Harmonic mean for the given data is 9.8 .

Example 2: Find Harmonic Mean if the Arithmetic Mean of the given data is 10 and the Geometric mean is 7.

Solution:

Given, 

Arithmetic Mean (AM) = 10
Geometric Mean (GM) = 7

We know that,

Harmonic Mean(HM) = (G.M)²/A.M

⇒ HM = 7²/10

⇒ HM = 49/10
⇒ HM = 4.9

Hence, the Harmonic mean from the given Arithmetic and geometric mean is 4.9

Example 3: Find the Geometric mean if the Arithmetic mean is 20 and the Harmonic mean is 15.

Solution:

Given,

Arithmetic Mean (A.M) = 20
Harmonic Mean (H.M) = 15

Geometric Mean(GM) = √(Arithmetic Mean × Harmonic Mean)

⇒ GM = √(20 × 15)

⇒ GM = √300
⇒ GM = 17.32

Hence, the Geometric mean from the given Arithmetic and Harmonic mean is 17.32

Example 4: Find the weighted harmonic mean for the given data.

Solution:

Weights(w)	x	1/x	w/x
1	20	0.05	0.05
2	30	0.03	0.06
3	10	0.1	0.3
2	15	0.06	0.12
∑w = 8			∑(w/x) = 0.53

Weight Harmonic Mean = ∑w / ∑(w/x)

⇒ Weight Harmonic Mean = 8/0.53
⇒ Weight Harmonic Mean = 15.09

The weighted Harmonic mean for the given data is 15.09 .

Example 5: Find the weighted harmonic mean for the given data.

Solution:

x	w	1/x	w/x
10	2	0.1	0.2
15	3	0.066	0.198
20	4	0.05	0.2
25	5	0.04	0.2
30	1	0.033	0.033
	∑w = 15		∑w/x = 0.831

Weighted Harmonic Mean = ∑w / ∑(w/x)

⇒ Weight Harmonic Mean = 15/0.831
⇒ Weight Harmonic Mean = 18.05

The weighted Harmonic mean for the given data is 18.05 .,