Harmonic Mean Function

A Harmonic Mean Function is a mean function that divides by the sum of the reciprocal of the members.

• Context:
  • range: Harmonic Mean Value.
  • It can be represented as [math]\displaystyle{ f(X) = \frac{n}{\frac{1}{a_i} + … + \frac{1}{a_n}} }[/math]
• Example(s):
  • [math]\displaystyle{ f \left( \frac{1}{4}, \frac{1}{2} \right) = \frac{2}{4+2} = \frac{1}{6} }[/math].
  • …
• Counter-Example(s):
  • Arithmetic Mean Function, [math]\displaystyle{ f \left( \frac{1}{4}, \frac{1}{2} \right) = \frac{3}{8} }[/math].
  • Geometric Mean Function.
• See: Statistic Function.

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References

2011

• http://en.wikipedia.org/wiki/Harmonic_mean
  • QUOTE:In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

    The harmonic mean H of the positive real numbers x₁, x₂, ..., x_n > 0 is defined to be :[math]\displaystyle{ H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}} = \frac{n \cdot \prod_{j=1}^n x_j }{ \sum_{i=1}^n \frac{\prod_{j=1}^n x_j}{x_i}}. }[/math]

    From the third formula in the above equation it is more apparent that the harmonic mean is related to the arithmetic mean and geometric mean.

    Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is :[math]\displaystyle{ \frac{3}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}} = \frac{1}{\frac{1}{3}(\frac{1}{1}+\frac{1}{2}+\frac{1}{4})} = \frac{12}{7}\,. }[/math]

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