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. First, the number of elements is averaged and divided by the sum of the reciprocals of the elements.
Harmonic mean formula is:
\({\rm{Harmonic \space Mean}} = \frac{n}{{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \ldots }} = \frac{n}{{\sum\limits_{i = n}^n {( {\frac{1}{{x_i }}} )} }}\)
where \(x_1, x_2, x_3, \ldots\) are \(n\) observations.
For a frequency distribution,
Harmonic mean formula is:
\({\rm{Harmonic \space Mean}} = \frac{n}{{\sum\limits_{i = n}^n {f ({\frac{1}{{x_i }}} )} }}\)
In the next section, we will solve some examples of the harmonic mean formula.
Solved 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
\({\frac{1}{x}}\)
\({f(\frac{1}{x})}\)
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 \(21.908\).