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, and is normally only used for positive arguments.

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

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

It is the reciprocal of the arithmetic mean of the reciprocals, and vice versa:

where the arithmetic mean is ${\textstyle A(x_{1},x_{2},\ldots ,x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.}$

The harmonic mean is a Schur-concave function, and is greater than or equal to the minimum of its arguments: for positive arguments, ${\displaystyle \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\leq n\min(x_{1}\ldots x_{n})}$. Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged). ^{[citation needed]}

The harmonic mean is also concave for positive arguments, an even stronger property than Schur-concavity.^{[citation needed]}