- Harmonic mean
mathematics, the harmonic mean (formerly 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 "x"1, "x"2, ..., "x""n" is defined to be
Equivalently, the harmonic mean is the reciprocal of the
arithmetic meanof 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 three Pythagorean means. For a given data set, the harmonic mean is always the least of the three, while the arithmetic meanis always the greatest of the three and the geometric meanis 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 of the
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 , ..., is associated to the dataset , ..., , the weighted harmonic mean is defined by:The harmonic mean is the special case where all weights are equal to 1.
In certain situations, especially many situations involving
rates and ratios, the harmonic mean provides the truest average. 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 the arithmetic meanof "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 a weighted harmonic meanor weighted arithmetic meanis needed.)
Similarly, if one connects two electrical
resistors 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 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 and , the harmonic mean can be written:
In this special case, the harmonic mean is related to the
arithmetic meanand the geometric meanby:
So , which means the geometric mean, for two numbers, is the geometric mean of the arithmetic mean and the harmonic mean.
* [http://mathworld.wolfram.com/HarmonicMean.html Harmonic Mean at MathWorld]
* [http://www.cut-the-knot.org/arithmetic/HarmonicMean.shtml Averages, Arithmetic and Harmonic Means] at
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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