Harmonic Mean is the type of mean that is used when we have to find the average rate of change, it is the mean calculated by taking the reciprocal values of the given value and then dividing the number of terms by the sum of the reciprocal values. The harmonic mean is one of the Pythagorean mean and the other two Pythagorean mean are,
- Arithmetic Mean
- Geometric Mean
These means tell us about various parameters of the data set.
Harmonic Mean also denoted as HM is the mean calculated by taking the reciprocal of the given set. In this article, we will learn about HM, its formula, examples, and others in detail.
What is Harmonic Mean?
Harmonic mean is defined as the average of the reciprocal values of the given values. That is for finding the harmonic mean of the given data set we first take the reciprocal of the data set and then divide the number of the data set by the sum of the reciprocal values. The value so obtained is called the Harmonic mean.
The harmonic mean is one of the measures of the central tendency which is generally used when we have to find the rate of the data set. It is used in the statistics and financial sector to explain various parameters of the data set.
Harmonic Mean Definition
Harmonic mean is one of the Pythagorean means other than Arithmetic Mean and Geometric Mean. The harmonic mean is always lower as compared to the geometric and arithmetic mean. The harmonic mean is calculated by dividing the number of the reciprocal by the sum of the reciprocal values.
The formula to calculate the harmonic mean is discussed in the image below:
Harmonic Mean Formula
The harmonic mean of the data set is calculated using the formula, Let x1, x2, x3, x4, ... xn 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, ..., xn 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/xn 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,...,wn and their values are x1, x2, x3, x4, ..., xn 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√(x1 . x2)
Taking the square of GM equality,
(GM)2 = x1 . x2
Now,
HM = 2 x1 . 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,
|
| We can calculate the Harmonic mean by dividing the number of values by the sum of the reciprocal values. | We can calculate the Geometric mean by taking the nth root of the product of all the data values. |
| The value of Harmonic Mean is the lowest among AM, GM, and HM | The value of Geometric Mean is lesser than the AM but greater than HM. |
The formula for HM of a, b is,
HM = (2ab)/(a + b)
| The formula for GM of a, b is,
GM = √(ab)
|
Example: Find the HM of 2, 4
HM = 2(2)(4)/(2+4) = 8/3
| Example: Find the GM of 2, 4
GM = √(2.4) = √(8) = 2√(2)
|
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,
|
| We can calculate the Harmonic mean by dividing the number of values by the sum of the reciprocal values. | We can calculate the Arithmetic mean by taking the sum of all the values and then dividing it by the number of the values. |
| The value of Harmonic Mean is the lowest among AM, GM, and HM | The value of Arithmetic Mean is the greatest among AM, GM, and HM |
The formula for HM of a, b is,
HM = (2ab)/(a + b)
| The formula for AM of a, b is,
AM = (a+b)/2
|
Example: Find the HM of 2, 4
HM = 2(2)(4)/(2+4) = 8/3
| Example: Find the AM of 2, 4
AM = (2+4)/2 = 3
|
Advantages and Disadvantages of Harmonic Mean
Harmonic mean is usually used in mathematics to find the average rate of change of the variables and the advantages and the disadvantages of the harmonic mean are,
Advantages of Harmonic Mean
Harmonic mean is used for finding the average value of the rate of changes of some specific type of data. Various advantages of the harmonic mean are,
It is completely based on observations and is very useful in averaging certain types of rates. Other merits of the harmonic mean are given below:
- The harmonic mean is rigidly defined and its value does not change so easily.
- The value of the harmonic mean is immune to simple fluctuation in the data.
- The harmonic mean is calculated after taking into consideration all the values of the data.
Disadvantages of Harmonic Mean
The harmonic mean is calculated only when all the values of the data set are given if there is any unknown value in the data set then the harmonic mean can not be calculated. Various disadvantages of the harmonic mean are,
- Calculating harmonic mean is complicated and lengthy as compared to other Pythagorous means, i.e. Arithmetic Mean and Geometric Mean.
- We can not calculate the harmonic mean if any of the values in the given series is zero.
- It is affected by the extreme values of the data set.
Uses of Harmonic Mean
Harmonic mean is widely used in finance, risk analysis, and insurance industries. Some of the uses of harmonic mean in real-life scenarios are,
- Harmonic mean is used in the Fibonacci series for finding various patterns.
- It is used to find the rate of change of various quantities, that include, displacement, velocity, etc.
- In financial calculations when average multiples are givens then harmonic mean is used.
Read More,
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:
|
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:
|
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 .
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