# HarmonicMean

HarmonicMean[data]

gives the harmonic mean of the values in data.

# Details

• For VectorQ data , the harmonic mean is given by .
• For MatrixQ data, HarmonicMean[{{x1,y1},{x2,y2},}] is equivalent to {HarmonicMean[{x1,x2,}],HarmonicMean[{y1,y2,}]}. »
• HarmonicMean handles both numerical and symbolic data.
• The data can have the following additional forms and interpretations:
•  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)

Harmonic mean of symbolic values:

Harmonic mean of columns of a matrix:

## Scope(12)

### Basic Uses(6)

Exact input yields exact output:

Approximate input yields approximate output:

Find the harmonic mean of WeightedData:

Find the harmonic mean of EventData:

Find the harmonic mean of a TimeSeries:

The harmonic mean depends only on the values:

Compute a weighted harmonic mean:

Find the harmonic mean of data involving quantities:

### Array Data(4)

HarmonicMean for a 2D matrix gives columnwise means:

Works with large arrays:

SparseArray data can be used just like dense arrays:

Find the harmonic mean of a QuantityArray:

### Image and Audio Data(2)

Channelwise harmonic mean value of an RGB image:

Harmonic mean intensity value of a grayscale image:

On audio objects, HarmonicMean works channelwise:

## Applications(1)

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

## Properties & Relations(6)

HarmonicMean is the inverse of Mean of the inverse of the data:

HarmonicMean is very sensible to values close to zero:

This agrees with the definition:

HarmonicMean is logarithmically related to GeometricMean for positive values:

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

Prove the inequality symbolically:

HarmonicMean[Range[n]] is inversely related to :

The harmonic mean of the shifted data is larger than the shifted harmonic mean of the original data (assuming the shift is a positive number):

## Possible Issues(1)

HarmonicMean will return 0 when the audio channels have a 0 in them; this is common for real-world audio samples:

Mean will clip channel values to values that can be represented with machine numbers:

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

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_harmonicmean, author="Wolfram Research", title="{HarmonicMean}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/HarmonicMean.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_harmonicmean, organization={Wolfram Research}, title={HarmonicMean}, year={2023}, url={https://reference.wolfram.com/language/ref/HarmonicMean.html}, note=[Accessed: 14-September-2024 ]}