What is "harmonic mean"

n. (context mathematics English) A type of measure of central tendency calculated as the reciprocal of the mean of the reciprocals, ie, $H\; =\; \{\; n\; over\; \{1\; over\; x\_1\}\; +\; \{1\; over\; x\_2\}\; +\; cdots\; \{1\; over\; x\_n\}\; \}$

n. the mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals. As a simple example, the harmonic mean of 1, 2, and 4 is $\frac{3}{\frac1{1}+\frac1{2}+\frac1{4}} = \frac1{\frac1{3}(\frac1{1}+\frac1{2}+\frac1{4})} = \frac{12}{7}\,.$

The harmonic mean H of the positive real numbers x, x, …, x is defined to be

$$H = \frac{n}{\frac1{x_1} + \frac1{x_2} + \cdots + \frac1{x_n}} = \frac{n}{\sum\limits_{i=1}^n \frac1{x_i}} = \frac{n\cdot \prod\limits_{j=1}^n x_j}{ \sum\limits_{i=1}^n \left\{\frac{1}{x_i}{\prod\limits_{j=1}^n x_j}\right\}}.$$

From the third formula in the above equation, it is more apparent that the harmonic mean is related to the arithmetic and geometric means. It is the reciprocal dual of the arithmetic mean for positive inputs:

1/H(1/x…1/x) = A(x…x)

The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments, min(x…x) ≤ H(x…x) ≤ nmin(x…x). Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged).