HarmonicMean

The reciprocal of the mean of the reciprocals.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageHarmonicMean✓↗

HarmonicMean(collection)$n \big/ \sum_{i} 1/x_i$ for the $n$ elements of $collection$.

Typed signatures

Type	Package	Applies
(collection<any>) -> number	combinatorics	always

Domain: Collections

Details

• See GeometricMean and Mean for the other Pythagorean means.

Examples

Basic #

$3 / (1 + \frac12 + \frac14) = \frac{12}{7}$

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In$\mathrm{HarmonicMean}(\bigl\lbrack1, 2, 4\bigr\rbrack)$Out$\frac{12}{7}$

$\frac{12}{5}$

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In$\mathrm{HarmonicMean}(\bigl\lbrack2, 3\bigr\rbrack)$Out$\frac{12}{5}$

The reciprocal of the mean of reciprocals, written out for symbols

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In$\mathrm{HarmonicMean}(\bigl\lbrack a, b, c, d\bigr\rbrack)$Out$\frac{4}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}}$

Applied per column of a matrix, each column giving an exact rational

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In$\mathrm{HarmonicMean}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack5, 10\bigr\rbrack, \bigl\lbrack5, 2\bigr\rbrack, \bigl\lbrack4, 8\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\frac{80}{33}, \frac{160}{49}\bigr\rbrack$

Scope #

An exact result for integers: $4/(1+\tfrac12+\tfrac13+\tfrac14) = 48/25$

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In$\mathrm{HarmonicMean}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\frac{48}{25}$

High-precision input gives an approximate value

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In$\mathrm{HarmonicMean}(N(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, 30))$Out$1.92$

Stays symbolic when the data is an unknown

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In$N(\mathrm{HarmonicMean}(\mathrm{ts}))$Out$\mathrm{HarmonicMean}(\mathrm{ts})$

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In$\mathrm{HarmonicMean}(\bigl\lbrack π, e, 2\bigr\rbrack)$Out$\frac{3}{\frac{1}{2}+\frac{1}{e}+\frac{1}{π}}$

Applications #

Driving equal distances at 40 and 60 averages 48, not 50

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In$\mathrm{HarmonicMean}(\bigl\lbrack40, 60\bigr\rbrack)$Out$48$

Properties #

Equal values are their own harmonic mean

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In$\mathrm{HarmonicMean}(\bigl\lbrack5, 5, 5\bigr\rbrack)$Out$5$

Reciprocal data turns the harmonic mean into $n$ over a sum of logs

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In$\mathrm{HarmonicMean}(\frac{1}{\ln(\bigl\lbrack a, b, c, d\bigr\rbrack)})$Out$\frac{4}{\ln(a)+\ln(b)+\ln(c)+\ln(d)}$

About

Examples	11
Checked against	Wolfram Language 11 agree
Crosswalk	Wolfram Language

See also: GeometricMean, Mean