# 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/a_{1} + 1/a_{2} + 1/a_{3} + 1/a_{4} + … +
1/a_{n})

Harmonic mean = H.M = n
/ Î£
(1/a_{i})

In the above equation,
n is the total number of observations in the given data, a_{1} 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/a_{1} + 1/a_{2} + 1/a_{3} + 1/a_{4} + … +
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/a_{1} + 1/a_{2} + 1/a_{3} + 1/a_{4} + … +
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:**