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 $f(x)={\frac {1}{x}}$ . For example, the harmonic mean of 1, 4, and 4 is

\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\,.

^ Course Archived 2022-07-11 at the Wayback Machine
^ Srivastava, U. K.; Shenoy, G. V.; Sharma, S. C. (1989). Quantitative Techniques for Managerial Decisions. New Age International. p. 63. ISBN 978-81-224-0189-9.
^ Jones, Alan (2018-10-09). Probability, Statistics and Other Frightening Stuff. Routledge. p. 42. ISBN 978-1-351-66138-6.

[1] Course Archived 2022-07-11 at the Wayback Machine

[2] Srivastava, U. K.; Shenoy, G. V.; Sharma, S. C. (1989). Quantitative Techniques for Managerial Decisions. New Age International. p. 63. ISBN 978-81-224-0189-9.

[3] Jones, Alan (2018-10-09). Probability, Statistics and Other Frightening Stuff. Routledge. p. 42. ISBN 978-1-351-66138-6.

[1]

[2]

[3]