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Encyclopedia > Harmonic mean

In 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. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... Look up Rate in Wiktionary, the free dictionary. ...


The harmonic mean H of the positive real numbers a1, ..., an is defined to be In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

That is, the harmonic mean of a group of terms is the reciprocal of the arithmetic mean of the terms' reciprocals. The reciprocal function: y = 1/x. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

Contents

Examples

In certain situations, the harmonic mean provides the correct notion of "average". For instance, if for half the distance of a trip you travel at 40 kilometres per hour and for the other half of the distance you travel at 60 kilometres per hour, then your average speed for the trip is given by the harmonic mean of 40 and 60, which is 48; that is, the total amount of time for the trip is the same as if you travelled the entire trip at 48 kilometres per hour. (Note however that if you had travelled for half the time at one speed and the other half at another, the arithmetic mean, in this case 50 kilometres per hour, would provide the correct notion of "average".) In mathematics, an average or central tendency of a set (list) of data refers to a measure of the middle of the data set. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...


Similarly, if in an electrical circuit you have two resistors connected in parallel, one with 40Ω and the other with 60Ω, then the average resistance of the two resistors is 48Ω; that is, the total resistance of the circuit is the same as it would be if each of the two resistors were replaced by a 48Ω resistor. (This is not to be confused with their equivalent resistance, 24Ω, which is the resistance needed for a single resistor to replace the two resistors at once.) Resistor symbols (non-European) Resistor symbols (Europe, IEC) A pack of resistors A resistor is a two-terminal electrical or electronic component that resists an electric current by producing a voltage drop between its terminals in accordance with Ohms law. ... The ohm (symbol: Ω) is the SI unit of electric resistance. ... This article or section does not adequately cite its references or sources. ...


In finance, the harmonic mean is used to calculate the average cost of shares purchased over a period of time. For example, an investor purchases $1000 worth of stock every month for three months. If the spot prices at execution time are as follows:

  • $8
  • $9
  • $10

Then the average price the investor paid is $8.926 per share. However, if the investor purchased 1000 shares per month, the arithmetic mean would be used.


Harmonic mean of two numbers

When dealing with just two numbers, an equivalent, sometimes more convenient, formula of their harmonic mean is given by:

In this case, their harmonic mean is related to their arithmetic mean, In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

and their geometric mean, The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ...

by

so

, i. e. the geometric mean is the geometric mean of the arithmetic mean and the harmonic mean.


Note that this result holds only in the case of just two numbers.


Relationship with other means

The harmonic mean is one of the 3 Pythagorean means and is always the lowest ot the 3, while the arithmetic mean is always the highest of the 3 and the geometric mean is always in between (it is actually the geometric mean of the other 2 Pythagorean means!) The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ...


It is the special case M − 1 of the power mean. A generalized mean, also known as power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means. ...


It is equivalent to a weighted arithmetic mean with each value's weight being the reciprocal of the value. In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted mean is calculated as Note that if all the weights are equal, the weighted mean is the same as the arithmetic 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.[1] In the speed example above for instance the arithmetic mean 50 is incorrect, and too big. Such an error was apparently made in a calculation of transport capacity of American ships during World War I. The arithmetic mean of the various ships' speed was used, resulting in a total capacity estimate which proved unattainable.[citation needed] Combatants Allied Powers: Russian Empire France British Empire Italy United States Central Powers: Austria-Hungary German Empire Ottoman Empire Bulgaria Commanders Nikolay II Aleksey Brusilov Georges Clemenceau Joseph Joffre Ferdinand Foch Robert Nivelle Herbert H. Asquith D. Lloyd George Sir Douglas Haig Sir John Jellicoe Victor Emmanuel III Luigi Cadorna...


See also

The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ... The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ... In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = { w1, w2, ..., wn} the weighted harmonic mean is calculated as Note that if all the weights are equal, the weighted harmonic mean is the same as the harmonic mean. ... Look up Rate in Wiktionary, the free dictionary. ... A generalized mean, also known as power mean or Hölder mean, is an abstraction of the Pythagorean means including arithmetic, geometric and harmonic means. ... In harmony, the diatessaron is a ratio of 4:3 (sesquitertium) between a pair of frequencies or, equivalently, a ratio of 3:4 between a pair of wavelengths. ... In harmony, the semiditonus is the ratio 6:5 (sesquiquintum) between a pair of frequencies or, equivalently, the ratio 5:6 between a pair of wavelengths. ...

References

  1. ^ *Statistical Analysis, Ya-lun Chou, Holt International, 1969, ISBN 0030730953

External links

  • The Rational Mean
  • Harmonic Mean at MathWorld
  • Averages, Arithmetic and Harmonic Means at cut-the-knot

  Results from FactBites:
 
Mean - Wikipedia, the free encyclopedia (906 words)
Sample mean is often used as an estimator of the central tendency such as the population mean.
The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not the same as the middle value (median), or most likely (mode).
The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).
  More results at FactBites »

 
 

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