Harmonic Mean

Elementary School

Definition

It is an average. Sometimes known as the subcontrary mean. In order to identify the harmonic mean in a set of numbers, you'll have to add the reciprocals of each of the numbers then divide it by n. Lastly, take the reciprocal of the results.

Worked examples

\(H = \frac{3}{\frac{1}{2} + \frac{1}{3} + \frac{1}{6}} = \frac{3}{\frac{3 + 2 + 1}{6}} = \frac{3}{1} = 3\)

For the set {2, 3, 6}: add reciprocals, divide by n=3, then take the reciprocal of the result.

\(H = \frac{2}{\frac{1}{60} + \frac{1}{40}} = \frac{2}{\frac{2 + 3}{120}} = \frac{2}{\frac{5}{120}} = \frac{240}{5} = 48\)

Harmonic mean of 60 and 40 is 48, useful for averaging rates like speed.

Common mistakes

\(H = \frac{\frac{1}{a} + \frac{1}{b}}{n}\) → \(H = \frac{n}{\frac{1}{a} + \frac{1}{b}}\) n goes in the numerator, not the denominator—you divide n by the sum of reciprocals, then take the reciprocal.
Harmonic mean of 4 and 8 is \(\frac{4 + 8}{2} = 6\) → \(H = \frac{2}{\frac{1}{4} + \frac{1}{8}} = \frac{16}{3} \approx 5.33\) That calculation is the arithmetic mean; harmonic mean uses reciprocals and is always smaller for different values.
Forgetting the final reciprocal step: \(H = \frac{1}{3} + \frac{1}{5} = \frac{8}{15}\) → \(H = \frac{2}{\frac{1}{3} + \frac{1}{5}} = \frac{2}{\frac{8}{15}} = \frac{30}{8} = 3.75\) After summing reciprocals and dividing by n, you must take the reciprocal of that quotient to get the harmonic mean.

Where you'll use it next

You'll use harmonic mean when averaging rates (like speeds or densities) in physics and statistics, in weighted mean problems, and later in advanced courses like machine learning for F-score and precision-recall metrics.

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