Harmonic Mean: Definition, Formulas and Applications

Harmonic mean is widely used in mathematics and statistics. It is one of the types of mean in the central tendency. The other types of the mean are geometric mean and arithmetic mean. These types are very essential in geometry and central tendency.

In statistics, this term is used to measure values or data in central tendency. While in mathematics, this term is used to find the divisor or multiplicative relationship among the fractions.

In this post, we will learn about the definition, formula, and examples of harmonic mean.

Harmonic Mean - Definition

Harmonic mean is stated as taking the reciprocal of the sum of the reciprocals of the given data values. In simple words, take the reciprocal of all the given values of the data and then add them and take the reciprocal of the added data.

The results of this kind of mean are always smaller as compared to the other two types of the mean. It is strictly defined and based on all the observations of the given set of data. It is also used to measure times and average rates.

Harmonic mean can also be stated in terms of arithmetic mean as the reciprocal (total data values) of the arithmetic mean of the reciprocal of the given set of observations.

For example, the harmonic mean of u and v is 2 / (1/a + 1/b).

Harmonic Mean - Formulas

Harmonic mean can be calculated by using the below equation.

Harmonic mean = H.M = n / (1/a₁ + 1/a₂ + 1/a₃ + 1/a₄ + … + 1/a_n)

Harmonic mean = H.M = n / Σ (1/a_i)

In the above equation, n is the total number of observations in the given data, a₁ is the first term of the data, and a_n is the last term of the data.

How to Calculate Harmonic Mean?

We can easily calculate the harmonic mean by using the formula of the harmonic mean. Let us take some examples to understand how to calculate the harmonic mean.

Example 1:

Find the harmonic mean of the given set of observations by using the formula.  

12, 14, 18, 22, 26, 28, 36

Solution:

Step 1: Take the given set of observations from the given statement.

12, 14, 18, 22, 26, 28, 36

Step 2: Now count the given observations.

n = 7

Step 3: Take the reciprocal of the given set of observations.

1/12, 1/14, 1/18, 1/22, 1/26, 1/28, 1/36

Step 4: Now find the result of each reciprocal value.

1/12 = 0.0833

1/14 = 0.0714

1/18 = 0.0556

1/22 = 0.0455

1/26 = 0.0385

1/28 = 0.0357

1/36 = 0.0278

Step 5: Now add the reciprocal results.

0.0833 + 0.0714 + 0.0556 + 0.0455 + 0.0385 + 0.0357 + 0.0278 = 0.3578

Step 6: Now, take the general formula of the harmonic mean.

Harmonic mean = H.M = n / (1/a₁ + 1/a₂ + 1/a₃ + 1/a₄ + … + 1/a_n)

Step 7: Now, put the values of the reciprocal results and total number of terms in the above equation.

Harmonic mean = H.M = 7 / (1/12 + 1/14 + 1/18 + 1/22 + 1/26 + 1/28 + 1/36)

Harmonic mean = H.M = 7 / (0.0833 + 0.0714 + 0.0556 + 0.0455 + 0.0385 + 0.0357 + 0.0278)

Harmonic mean = H.M = 7 / 0.3578

Harmonic mean = H.M = 19.5681

Hence, the harmonic mean of 12, 14, 18, 22, 26, 28, 36 is 19.5681.

To avoid such large calculations, you can use harmonic mean calculator in which you have to put your question and the result will be in front of you after clicking the Calculate button.

 

Example 2:

Find the harmonic mean of the given set of observations by using the formula.

11, 17, 28, 32, 36, 38, 46, 56, 60

Solution:

Step 1: Take the given set of observations from the given statement.

11, 17, 28, 32, 36, 38, 46, 56, 60

Step 2: Now count the given observations.

n = 9

Step 3: Take the reciprocal of the given set of observations.

1/11, 1/17, 1/28, 1/32, 1/36, 1/38, 1/46, 1/56, 1/60

Step 4: Now find the result of each reciprocal value.

1/11 = 0.0909

1/17 = 0.0588

1/28 = 0.0357

1/32 = 0.0313

1/36 = 0.0278

1/38 = 0.0263

1/46 = 0.0217

1/56 = 0.0179

1/60 = 0.0167

Step 5: Now, add the reciprocal results.

0.0909 + 0.0588 + 0.0357 + 0.0313 + 0.0278 + 0.0263 + 0.0217 + 0.0179 + 0.0167 = 0.3271

Step 6: Now, take the general formula of the harmonic mean.

Harmonic mean = H.M = n / (1/a₁ + 1/a₂ + 1/a₃ + 1/a₄ + … + 1/a_n)

Step 7: Now, put the values of the reciprocal results and total numbers of terms in the above equation.

Harmonic mean = H.M = 9 / (1/11 + 1/17 + 1/28 + 1/32 + 1/36 + 1/38 + 1/46 + 1/56 + 1/60)

Harmonic mean = H.M = 9 / (0.0909 + 0.0588 + 0.0357 + 0.0313 + 0.0278 + 0.0263 + 0.0217 + 0.0179 + 0.0167)

Harmonic mean = H.M = 9 / 0.3271

Harmonic mean = H.M = 27.5184

Hence, the harmonic means of 11, 17, 28, 32, 36, 38, 46, 56, 60 is 27.5184.

Applications of the Harmonic Mean

There are many uses and applications of the harmonic mean. Let us discuss some of them.

·         The harmonic mean is used to calculate the multiplicative and divisor relationship among the fractions without taking the LCM of the denominators for making the same denominator.

·         The harmonic mean is used to determine the pattern of the Fibonacci sequence.

·         To measure the average multiples in finance, the harmonic mean is used.

·         It is also very essential to calculate the quantities that are expressed in ratios like speed.

Summary

The above-discussed type of mean is very essential for solving different kinds of problems. By following the above examples and formula, you can easily solve any problem related to harmonic mean.

Related Topics:

Mean, Median, Mode

Introduction to Limits