HarmonicMean

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 »

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Harmonic mean of symbolic values:

Harmonic mean of columns of a matrix:

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:

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:

Channelwise harmonic mean value of an RGB image:

Harmonic mean intensity value of a grayscale image:

On audio objects, HarmonicMean works channelwise:

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

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 HarmonicNumber[n]:

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):

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:

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