Harmonic mean

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means. It is sometimes used for ratios and rates such as speeds, and is normally used for positive arguments only. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals of the numbers, that is, the generalized f-mean with f ( x ) = 1 x {\displaystyle f(x)={\frac {1}{x}}} . For example, the harmonic mean of 1, 4, and 4 is ( 1 − 1 + 4 − 1 + 4 − 1 3 ) − 1 = 3 1 1 + 1 4 + 1 4 = 3 1.5 = 2 . {\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\,.}