Definition:Harmonic Mean
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This page is about harmonic mean in the context of algebra. For other uses, see mean.
Definition
Let $x_1, x_2, \ldots, x_n \in \R$ be real numbers which are all strictly positive.
The harmonic mean $H_n$ of $x_1, x_2, \ldots, x_n$ is defined as:
- $\ds \dfrac 1 {H_n} := \frac 1 n \paren {\sum_{k \mathop = 1}^n \frac 1 {x_k} }$
That is, to find the harmonic mean of a set of $n$ numbers, take the reciprocal of the arithmetic mean of their reciprocals.
Also see
- Results about the harmonic mean can be found here.
Sources
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of Mathematical Functions ... (previous) ... (next): $3$: Elementary Analytic Methods: $3.1$ Binomial Theorem etc.: Harmonic Mean of $n$ quantities $H$: $3.1.13$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 36$: Inequalities: Inequalities involving Arithmetic, Geometric and Harmonic Means: $36.7$
- 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 1$: Real Numbers: Exercise $\S 1.12 \ (5)$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): harmonic mean
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): harmonic mean
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): mean: 3.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): harmonic mean
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): mean: 3.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): harmonic mean
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): mean
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): harmonic mean
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): mean