Harmonic mean

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired. The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is ${\displaystyle {\frac {1}{{\frac {1}{3}}({\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}})}}={\frac {3}{{\frac {1}{1}}+{\frac {1}{2}}+{\frac {1}{4}}}}={\frac {12}{7}}\,.}$

The harmonic mean H of the positive real numbers ${\displaystyle x_{1},x_{2},\ldots ,x_{n}}$ is defined to be

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