# GeometricMean

GeometricMean[data]

gives the geometric mean of the values in data.

# Details

• For VectorQ data {x1,x2,,xn}, the geometric mean is given by .
• GeometricMean[{{x1,y1,},{x2,y2,},}] gives {GeometricMean[{x1,x2,}],GeometricMean[{y1,y2,}]}. »
• For ArrayQ data, the geometric mean estimate is equivalent to ArrayReduce[GeometricMean,data,1]. »
• GeometricMean handles both numerical and symbolic data.
• The data can have the following additional forms and interpretations:
•  Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData weighted mean, based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channel's values or grayscale intensity value » Audio amplitude values of all channels »

# Examples

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## Basic Examples(2)

Geometric mean of a list:

Geometric mean of columns of a matrix:

## Scope(13)

### Basic Uses(6)

Exact input yields exact output:

Approximate input yields approximate output:

Find the geometric mean of WeightedData:

Find the geometric mean of EventData:

Find the geometric mean of a TimeSeries:

The geometric mean depends only on the values:

Compute a weighted geometric mean:

Find the geometric mean of data involving quantities:

### Array Data(5)

GeometricMean for a matrix gives columnwise means:

Mean for a tensor works across the first index: »

Works with large arrays:

When the input is an Association, GeometricMean works on its values:

SparseArray data can be used just like dense arrays:

Find the geometric mean of a QuantityArray:

### Image and Audio Data(2)

Channelwise geometric mean value of an RGB image:

Geometric mean intensity value of a grayscale image:

On audio objects, GeometricMean works channelwise:

## Applications(1)

Find the geometric mean for the heights of children in a class:

## Properties & Relations(3)

GeometricMean is logarithmically related to Mean for positive values:

GeometricMean is logarithmically related to HarmonicMean for positive values:

For positive data, HarmonicMean[d]GeometricMean[d]Mean[d]:

Prove the inequality symbolically:

## Possible Issues(1)

GeometricMean may return complex values when data contains negative values:

For different sample realizations, the geometric mean is real:

Wolfram Research (2007), GeometricMean, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricMean.html (updated 2023).

#### Text

Wolfram Research (2007), GeometricMean, Wolfram Language function, https://reference.wolfram.com/language/ref/GeometricMean.html (updated 2023).

#### CMS

Wolfram Language. 2007. "GeometricMean." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/GeometricMean.html.

#### APA

Wolfram Language. (2007). GeometricMean. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeometricMean.html

#### BibTeX

@misc{reference.wolfram_2023_geometricmean, author="Wolfram Research", title="{GeometricMean}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/GeometricMean.html}", note=[Accessed: 30-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_geometricmean, organization={Wolfram Research}, title={GeometricMean}, year={2023}, url={https://reference.wolfram.com/language/ref/GeometricMean.html}, note=[Accessed: 30-September-2023 ]}