- Harmonic mean
In

mathematics , the**harmonic mean**(formerly sometimes called the**subcontrary mean**) is one of several kinds ofaverage . Typically, it is appropriate for situations when the average of rates is desired.The harmonic mean "H" of the positive

real number s "x"_{1}, "x"_{2}, ..., "x"_{"n"}is defined to be:$H\; =\; frac\{n\}\{frac\{1\}\{x\_1\}\; +\; frac\{1\}\{x\_2\}\; +\; cdots\; +\; frac\{1\}\{x\_n\; =\; frac\{n\}\{sum\_\{i=1\}^n\; 1/x\_i\},\; qquad\; x\_i\; >\; 0\; ext\{\; for\; all\; \}\; i.$

Equivalently, the harmonic mean is the reciprocal of the

arithmetic mean of the reciprocals.**Relationship with other means**[

Pythagorean means (of two numbers only). Harmonic mean denoted by "HM" in purple colour.] The harmonic mean is one of the threePythagorean means . For a given data set, the harmonic mean is always the least of the three, while thearithmetic mean is always the greatest of the three and thegeometric mean is always in between. When the data values to be averaged are equal to each other, the harmonic mean is equal to both the geometric mean and the arithmetic mean. For example, if the values are 2 and 2, the harmonic mean, geometric mean, and arithmetic mean are all equal (in this case, 2).It is the special case $M\_\{-\; 1\}$ of the

power mean .Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.

The arithmetic mean is often incorrectly used in places calling for the harmonic mean. [

**"Statistical Analysis", Ya-lun Chou, Holt International, 1969, ISBN 0030730953*] In the speed example below for instance the arithmetic mean 50 is incorrect, and too big.**Weighted harmonic mean**If a set of weights $w\_1$, ..., $w\_n$ is associated to the dataset $x\_1$, ..., $x\_n$, the

**weighted harmonic mean**is defined by:$sum\_\{i=1\}^n\; w\_i\; igg/\; sum\_\{i=1\}^n\; frac\{w\_i\}\{x\_i\}.$The harmonic mean is the special case where all weights are equal to 1.**Examples****In physics**In certain situations, especially many situations involving

rate s andratio s, the harmonic mean provides the truestaverage . For instance, if a vehicle travels a certain distance at a speed "x" (e.g. 60 kilometres per hour) and then the same distance again at a speed "y" (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of "x" and "y" (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of "time" at a speed "x" and then the same amount of time at a speed "y", then its average speed is thearithmetic mean of "x" and "y", which in the above example is 50 kilometres per hour. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same "distance", then the average speed is the "harmonic" mean of all the sub-trip speeds, and if each sub-trip takes the same amount of "time", then the average speed is the "arithmetic" mean of all the sub-trip speeds. (If neither is the case, then aweighted harmonic mean orweighted arithmetic mean is needed.)Similarly, if one connects two electrical

resistor s in parallel, one having resistance "x" (e.g. 60Ω) and one having resistance "y" (e.g. 40Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of "x" and "y" (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of "x" and "y" (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.**In other sciences**An interesting consequence arises from basic algebra in problems of working together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps $frac$6} cdot {4 6} + {4, which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4.

**Harmonic mean of two numbers**For the special case of just two numbers $x\_1$ and $x\_2$, the harmonic mean can be written:$H\; =\; frac\{2\; x\_1\; x\_2\}\{x\_1\; +\; x\_2\}.$

In this special case, the harmonic mean is related to the

arithmetic mean $A\; =\; (x\_1\; +\; x\_2)/2$and thegeometric mean $G\; =\; sqrt\{x\_1\; x\_2\},$ by:$H\; =\; frac\; \{G^2\}\; \{A\}.$So $G\; =\; sqrt\{A\; H\}$, which means the geometric mean, for two numbers, is the geometric mean of the arithmetic mean and the harmonic mean.

**ee also***

Harmonic number

*Rate

*Generalized mean **References****External links*** [

*http://mathworld.wolfram.com/HarmonicMean.html Harmonic Mean at MathWorld*]

* [*http://www.cut-the-knot.org/arithmetic/HarmonicMean.shtml Averages, Arithmetic and Harmonic Means*] atcut-the-knot

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### См. также в других словарях:

**harmonic mean**— n. a number associated with a set of numbers, that is equal to the number of numbers divided by the sum of the reciprocals of the numbers (h = n ÷ (1/a + 1/b)) (Ex.: for 1/ 2, 1/ 3, and 1/ 4, h = 3 ÷ (2 + 3 + 4) = 1/ 3 or for 1/ 2 and 1/ 3, h = 2 … English World dictionary**harmonic mean**— noun the mean of n numbers expressed as the reciprocal of the arithmetic mean of the reciprocals of the numbers • Topics: ↑statistics • Hypernyms: ↑mean, ↑mean value * * * noun 1. : the reciprocal of the arithmetic mean of the reciprocals of two… … Useful english dictionary**harmonic mean**— harmoninis vidurkis statusas T sritis Standartizacija ir metrologija apibrėžtis Apibrėžtį žr. priede. priedas( ai) Grafinis formatas atitikmenys: angl. harmonic average; harmonic mean vok. harmonisches Mittel, n rus. среднее гармоническое, n… … Penkiakalbis aiškinamasis metrologijos terminų žodynas**harmonic mean**— harmoninis vidurkis statusas T sritis fizika atitikmenys: angl. harmonic average; harmonic mean vok. harmonisches Mittel, n rus. среднее гармоническое, n pranc. moyenne harmonique, f … Fizikos terminų žodynas**harmonic mean**— reciprocal of the mean of the reciprocals of the individual values in a given set; e.g., for the set [10, 40, 60] the harmonic mean is 1 Ñ‡ [ … Medical dictionary**harmonic mean**— harmon′ic mean′ n. sta the statistical mean obtained by taking the reciprocal of the arithmetic mean of the reciprocals of a set of nonzero numbers • Etymology: 1880–85 … From formal English to slang**harmonic mean**— Statistics. the mean obtained by taking the reciprocal of the arithmetic mean of the reciprocals of a set of nonzero numbers. [1880 85] * * * … Universalium**harmonic mean**— noun Date: 1856 the reciprocal of the arithmetic mean of the reciprocals of a finite set of numbers … New Collegiate Dictionary**harmonic mean**— Reciprocal of the arithmatic mean … Dictionary of invertebrate zoology**harmonic mean**— noun A type of measure of central tendency calculated as the reciprocal of the mean of the reciprocals, ie … Wiktionary