GeometricMean

The n-th root of the product of n values.

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GeometricMean(collection)$\sqrt[n]{x_1 x_2 \cdots x_n}$ for the $n$ elements of $collection$.

Typed signatures

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

Domain: Collections

Details

• See HarmonicMean and Mean for the other Pythagorean means.

Examples

1 more kept as data, checked against the oracles

Basic #

$\sqrt{2 \cdot 8} = 4$

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In$\mathrm{GeometricMean}(\bigl\lbrack2, 8\bigr\rbrack)$Out$4$

$\sqrt[3]{1 \cdot 2 \cdot 4} = 2$

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In$\mathrm{GeometricMean}(\bigl\lbrack1, 2, 4\bigr\rbrack)$Out$2$

For symbols this is the fourth root of the product, $\sqrt[4]{abcd}$

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In$\mathrm{GeometricMean}(\bigl\lbrack a, b, c, d\bigr\rbrack)$Out$\sqrt[4]{abcd}$

A matrix is averaged column by column, each exact root left unevaluated

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In$\mathrm{GeometricMean}(\bigl\lbrack\bigl\lbrack5, 10\bigr\rbrack, \bigl\lbrack2, 1\bigr\rbrack, \bigl\lbrack4, 3\bigr\rbrack, \bigl\lbrack12, 15\bigr\rbrack\bigr\rbrack)$Out$\bigl\lbrack\sqrt[4]{480}, \sqrt[4]{450}\bigr\rbrack$

Scope #

An exact irrational result

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In$\mathrm{GeometricMean}(\bigl\lbrack2, 3\bigr\rbrack)$Out$\sqrt{6}$

$\sqrt[3]{6}$

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In$\mathrm{GeometricMean}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\sqrt[3]{6}$

Symbolic data

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In$\mathrm{GeometricMean}(\bigl\lbrack a, b\bigr\rbrack)$Out$\sqrt{ab}$

Exact integers keep an exact radical, $\sqrt[4]{24}$

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In$\mathrm{GeometricMean}(\bigl\lbrack1, 2, 3, 4\bigr\rbrack)$Out$\sqrt[4]{24}$

A 30-digit numeric list gives a 30-digit result

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In$\mathrm{GeometricMean}(N(\bigl\lbrack1, 2, 3, 4\bigr\rbrack, 30))$Out$2.213\,363\,839\,400\,643\,184\,82$

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In$\mathrm{GeometricMean}(\bigl\lbrack π, e, 2\bigr\rbrack)$Out$\sqrt[3]{2e π}$

Properties #

The log of a geometric mean is the arithmetic mean of the logs

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

The same identity read through a reciprocal

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

About

Examples	13
Checked against	Wolfram Language 13 agree
Crosswalk	Wolfram Language

See also: HarmonicMean, Mean