Harmonic Mean
2026-02-28 00:46 Diff
https://brightchamps.com/en-us/math/data/harmonic-mean

The harmonic mean is used for analyzing data involving rates, ratios, or quantities such as speed, time, and financial multiples. It plays a vital role in the fields of physics, statistics, and finance. The harmonic mean is very helpful when a data set’s smaller values have a greater impact or significance. Suppose we have a set of observations such as\(x_1, x_2, x_3, \dots, x_n \) Then, the reciprocal terms of this data set will be\(\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \dots, \frac{1}{x_n} \) So, the formula for the harmonic mean is:

\( HM = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \frac{1}{x_3} + \dots + \frac{1}{x_n}} \)

Here, n is the number of terms in the given data set. 


\(x1, x2, x3… xn\) are the values in the given data set.  


In the formula, the total number of terms is divided by the sum of the reciprocal of each number. 


For a better understanding, take a look at the given example.

Imagine we have a sequence given by 2, 6, 10, 14. The difference between each term is 4, creating an arithmetic progression. To calculate the harmonic mean, first, we can take the reciprocals of these terms.

\(\frac{1}{2}, \frac{1}{6}, \frac{1}{10}, \frac{1}{14} \)

This creates a harmonic progression. Next, we can divide the total number of terms, i.e., 4 by the sum of the reciprocal terms:

\(HM = \frac{4}{\frac{1}{2} + \frac{1}{6} + \frac{1}{10} + \frac{1}{14}} \)

\( \frac{1}{2} + \frac{1}{6} + \frac{1}{10} + \frac{1}{14} = 0.5 + 0.1667 + 0.1 + 0.0714 = 0.8381 \)

\(HM = n / ( 1/x1 + 1/x2 + 1/ x3 +…1/xn)\)


\(HM = 4 / 0.8381 ≈ 4.773\)


So, the harmonic mean of the sequence is approximately 4.77.