Harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

It is the most appropriate average for ratios and rates such as speeds,^[1][2] and is normally only used for positive arguments.^[3]

The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with ${\displaystyle f(x)={\frac {1}{x}}}$. For example, the harmonic mean of 1, 4, and 4 is

${\displaystyle \left({\frac {1^{-1}+4^{-1}+4^{-1}}{3}}\right)^{-1}={\frac {3}{{\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{4}}}}={\frac {3}{1.5}}=2\,.}$