# Harmonic Mean Formula

The harmonic mean formula is a type of average calculator that is calculated by dividing the number of values by the sum of the reciprocals of each value in the data series or in other words the harmonic mean is the reciprocal of the average of the reciprocals. Please note that the harmonic mean, arithmetic mean, and geometric mean are the three Pythagorean means and the harmonic mean always shows the lowest value among the Pythagorean means.

## What Is Harmonic Mean Formula?

The harmonic mean formula is used to calculate the average of a set of numbers. The number of elements is averaged and divided by the sum of the reciprocals of the elements.

### Harmonic Mean Formula

Harmonic mean formula is given as

### \({\rm{Harmonic \space Mean}} = \frac{n}{{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots }} = \frac{n}{{\sum\limits_{i = 1}^n {( {\frac{1}{{x_i }}} )} }}\)

where \(x_1, x_2, x_3, \ldots\) are \(n\) observations.

For a frequency distribution, the harmonic mean formula is:

### \({\rm{Harmonic \space Mean}} = \frac{n}{{\sum\limits_{i = 1}^n {f ({\frac{1}{{x_i }}} )} }}\)

## Applications of Harmonic Mean Formula

A few common applications of the harmonic mean formula are given below

- used in calculating the average under certain conditions.
- computing Fibonacci sequences.
- used in finance, specifically to calculate average multiples.

In the next section, we will solve some examples of the harmonic mean formula.

**Break down tough concepts through simple visuals.**

## Examples Using Harmonic Mean Formula

**Example 1:** From the given data 5, 10, 17, 24, 30. Calculate the harmonic mean.

**Solution:**

There are a total of 5 observations.

Hence, \(\begin{array}{c} {\rm{Harmonic \space Mean}} = \frac{5}{{\frac{1}{5} + \frac{1}{{10}} + \frac{1}{{17}} + + \frac{1}{{24}} + \frac{1}{{30}}}} \\ = \frac{5}{{0.2 + 0.1 + 0.0588 + 0.0417 + 0.0333}} \\ = \frac{5}{{0.4335}} \\ = 11.534 \\ \end{array}\)

**Answer:** **Harmonic mean for the given data is 11.534.**

**Example 2:** The number of tomatoes per plant is given below. Calculate the harmonic mean.

Number of tomatoes per plant | 20 | 21 | 22 | 23 | 24 | 25 |

Number of plants | 4 | 2 | 7 | 1 | 3 | 1 |

**Solution:**

Number of tomatoes per plant \(x\) |
Number of plants |
\({\dfrac{1}{x}}\) |
\(f(\dfrac{1}{x})\) |

20 | 4 | 0.05 | 0.2 |

21 | 2 | 0.0476 | 0.0952 |

22 | 7 | 0.0454 | 0.3178 |

23 | 1 | 0.0435 | 0.0435 |

24 | 3 | 0.0417 | 0.1251 |

25 | 1 | 0.04 | 0.04 |

n=18 |
0.8216 |

Hence, \({\rm{Harmonic \space Mean}} = \frac{n}{{\sum\limits_{i = n}^n {f ({\frac{1}{{x_i }}} )} }}={\frac{18}{{0.8216 }}}=21.908 \)

**Answer: Harmonic mean for the given data is 21.908**

**Example 3**: Calculate the harmonic mean from the given data set: 2, 5, 6, 8, 10.

**Solution:**

There are a total of 5 observations.

Hence, \(\begin{array}{c} {\rm{Harmonic \space Mean}} = \frac{5}{{\frac{1}{2} + \frac{1}{{5}} + \frac{1}{{6}} + + \frac{1}{{8}} + \frac{1}{{10}}}} \\ = \frac{5}{{0.5 + 0.2 + 0.166 + 0.125 + 0.1}} \\ = \frac{5}{{1.091}} \\ = 4.5829 \\ \end{array}\)

**Answer:** **Harmonic mean for the given data is 0.829.**

## FAQs on Harmonic Mean Formula

### What Is the Harmonic Mean Formula in Statistics?

The harmonic mean formula is a formula to calculate the average of a set of numbers. The harmonic mean formula is given as \({\rm{Harmonic \space Mean}} = \frac{n}{{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots }} = \frac{n}{{\sum\limits_{i = 1}^n {( {\frac{1}{{x_i }}} )} }}\), where \(x_1, x_2, x_3, \ldots\) are \(n\) observations.

### What Is the Harmonic Mean Formula for Grouped Data?

For a grouped data, the harmonic mean formula H.M = \(\frac{f_{1}+f_{2}+f_{3}+…+f_{n}}{\frac{f_{1}}{x_{1}}+\frac{f_{2}}{x_{2}}+\frac{f_{3}}{x_{3}}+…+\frac{f_{n}}{x_{n}}}\) = \(\frac{\sum f}{\sum (\frac{f}x{})}\) for the individual values \(x_1\), \(x_2\), \(x_3\), …., \(x_n\)_{ }and frequencies \(f_1\), \(f_2\), \(f_3\), ….., \(f_n\)_{. }

### How To Use Harmonic Mean Formula?

For the given values, say a, b, c, d, …

- Step 1: Find the reciprocal of each value as 1/a, 1/b, 1/c, 1/d, …
- Step 2: Find the average of reciprocals.
- Step 3: Compute the reciprocal of the average so obtained.
- Harmonic mean = n/[(1/a)+1/b + 1/c +1/d........]

### What Is the Harmonic Mean Formula for a and b?

Harmonic Mean formula is a kind of average formula only. Thus, the harmonic mean between two numbers, say a and b is calculated as, H = 2/ (1/a + 1/b)