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# Topic: Harmonic mean

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 Mean - Wikipedia, the free encyclopedia 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). en.wikipedia.org /wiki/Mean   (906 words)

 Harmonic mean: Definition and Links by Encyclopedian.com - All about Harmonic mean   (Site not responding. Last check: 2007-10-19) In mathematics, the harmonic mean is one of several methods of calculating an average. The harmonic mean is never larger than the geometric mean or the arithmetic mean (see generalized mean). In certain situations, the harmonic mean provides the correct notion of "average". www.encyclopedian.com /ha/Harmonic-mean.html   (186 words)

 Harmonic mean -- Facts, Info, and Encyclopedia article   (Site not responding. Last check: 2007-10-19) In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the harmonic mean is one of several methods of calculating an (A statistic describing the location of a distribution) average. The harmonic mean is never larger than the (The mean of n numbers expressed as the n-th root of their product) geometric mean or the (The sum of the values of a random variable divided by the number of values) arithmetic mean (see (Click link for more info and facts about generalized mean) generalized mean). In certain situations, the harmonic mean provides the correct notion of " (A statistic describing the location of a distribution) average". www.absoluteastronomy.com /encyclopedia/h/ha/harmonic_mean.htm   (464 words)

 Estimation of Harmonic Mean of Lognormal Variable   (Site not responding. Last check: 2007-10-19) The harmonic mean has numerous engineering applications including characterization of the large-scale effective permeability in layered porous media, characterization of petrochemical properties of heterogeneous media, and the design of declining rate filter beds. The sampling properties of various estimators of the harmonic mean are derived and compared for observations arising from a lognormal distribution. We document that the moment estimator of the harmonic mean exhibits significant upward bias and large root mean-square error, particularly for large skews. www.pubs.asce.org /WWWdisplay.cgi?0000063   (208 words)

 Harmonic mean - Wikipédia Dina matematik, the harmonic mean is one of several methods of calculating an average. Similarly, if in an electrical circuit you have two resistors connected in parallel, one with 40 ohms and the other with 60 ohms, then the average resistance is 24 ohms, which is half of the harmonic mean, as there are as many paths for current to travel as there are resistors (i.e. Typically the harmonic mean is appropriate for situations when the average of a rate is desired. su.wikipedia.org /wiki/Harmonic_mean   (435 words)

 Encyclopedia: Harmonic mean The geometric mean of a set of positive data is defined as the product of all the members of the set, raised to a power equal to the reciprocal of the number of members. In descriptive statistics, when the scores are available in the form of a regular frequency distribution, the harmonic means is computed as follows : Descriptive statistics is a branch of statistics that denotes any of the many techniques used to summarize a set of data. Categories: 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. www.nationmaster.com /encyclopedia/Harmonic-mean   (792 words)

 Geometric Mean and Harmonic Mean Not to worry: the mean is the appropriate single-point forecast 95+% of evaluation uses. The harmonic mean is a better "average" when the numbers are defined in relation to some unit. In petroleum engineering, the harmonic mean is sometimes the better "average" for vertical permeability with horizontally-layered bedding. www.maxvalue.com /tip104.htm   (488 words)

 PlanetMath: mean Loosely speaking, a mean is a way to describe a collection of numbers such that the mean in some sense describe the ``average'' entry of these numbers. Pythagoras identified three types of means: the arithmetic mean, the geometric mean, and the harmonic mean. This is version 7 of mean, born on 2002-06-04, modified 2005-03-01. planetmath.org /encyclopedia/Mean3.html   (179 words)

 PlanetMath: arithmetic mean   (Site not responding. Last check: 2007-10-19) The arithmetic mean is what is commonly called the average of the numbers. See Also: geometric mean, harmonic mean, arithmetic-geometric-harmonic means inequality, general means inequality, weighted power mean, power mean, geometric distribution, root-mean-square, proof of general means inequality, proof of arithmetic-geometric-harmonic means inequality, derivation of geometric mean as the limit of the power mean, mean, a prime theorem of a convergent sequence This is version 4 of arithmetic mean, born on 2001-10-20, modified 2004-04-15. planetmath.org /encyclopedia/Mean.html   (102 words)

 Math Forum - Ask Dr. Math   (Site not responding. Last check: 2007-10-19) Iamblichus says that the harmonic mean "was then called subcontrary, but which was renamed harmonic by the circle of Archytas and Hippasus, because it seemed to furnish harmonius and tuneful ratios." There are lots of other neat properties of means. s is half the harmonic mean of the base of the triangle and the altitude of the triangle on the base. For example, if you want the harmonic mean of 10 and 20, you first take 1/10 and 1/20, find their average, which is 3/40, and then take the reciprocal of that, 40/3. mathforum.org /library/drmath/view/57565.html   (681 words)

 [No title] Prove that the reciprocal of the harmonic mean of x and y is the arithmetic mean of the reciprocals of x and y. Therefore, the reciprocal of the harmonic mean is equal to the arithmetic mean. Write the harmonic mean of x and y as the quotient of two polynomials in x and y. www.arches.uga.edu /~ann/4500Assignment12.doc   (215 words)

 Essay3 In a trapezoid with bases of lengths "a" and "b," the harmonic mean, "H," is the length of the segment that is parallel to the bases and that also passes through the intersection of the diagonals of the trapezoid. In a trapezoid with bases of lengths "a" and "b," the heronian mean, "h," is the length of the segment that is parallel to the bases and that also is 1/ 3 the way from the arithmetic mean to the geometric mean. In a trapezoid with bases of lengths "a" and "b," the contraharmoic mean, "c," is the length of the segment that is parallel to the bases and that also is as far below the arithmetic mean as the harmonic mean is above the arithmetic mean. jwilson.coe.uga.edu /EMT668/EMAT6680.2000/Umberger/EMAT6690smu/Essay3smu/Essay3smu.html   (722 words)

 Weighted harmonic mean - Wikipedia, the free encyclopedia Note that if all the weights are equal, the weighted harmonic mean is the same as the harmonic mean. Probably the best known weighted mean is the weighted arithmetic mean, usually simply called the weighted mean. Another example of a weighted mean is the weighted geometric mean. www.wikipedia.org /wiki/Weighted_harmonic_mean   (90 words)

 Harmony and Proportion: Palladio's Room Proportions: The Harmonic Mean In other words the Harmonic Mean is the mean exceeding one extreme, and being exceeded by the other, by the same fraction of the extremes. The mean, 8, exceeds the smaller extreme, 6, by a third of the smaller extreme; 2, just as it (the mean) is itself exceeded by the same fraction (a third) of the larger extreme, 12, which is 4. Thus 12 times 6 gives 72, which is then divided by the arithmetical mean, 9, to give the answer 8 which is the harmonic mean; the height of the room. www.aboutscotland.com /harmony/prop6.html   (501 words)

 Mean article - Mean statistics average arithmetic mean geometric mean harmonic mean - What-Means.com   (Site not responding. Last check: 2007-10-19) Note that not every probability distribution has a defined mean or variance - see the Cauchy distribution for an example. The geometric mean is an average which is useful for sets of numbers which are interpreted according to their product and not their sum (as is the case with the arithmetic mean). The generalized mean is an abstraction of the Arithmetic, Geometric and Harmonic Means. www.what-means.com /encyclopedia/Mean   (748 words)

 The Super-Symmetric Mean   (Site not responding. Last check: 2007-10-19) The Holder mean M_k() is defined by the above formula with f(x)=x^k, from which it follows that M_1() is the Arithmetic mean, M_-1() is the Harmonic mean, M_2() is the root-sum-square, and the limit of M_k() as k goes to infinity is the Geometric mean. However, these means are all just (in a sense) weighted version of a single prototype, and don't seem to capture the essence of the desired generalization of the AGM. Again the mean polynomials match the value and the first and second derivatives of the function f(x)=(x-3)(x-7)(x-15) respectively at x=0. www.mathpages.com /home/kmath461.htm   (768 words)

 Various Averages and Means OT is the arithmetic mean of a and b, TF is the harmonic mean of a and b, AR is the quadratic mean of a and b. www.cut-the-knot.com /arithmetic/Means.shtml   (1045 words)

 PlanetMath: harmonic mean See Also: arithmetic mean, general means inequality, weighted power mean, power mean, arithmetic-geometric-harmonic means inequality, root-mean-square, proof of general means inequality, proof of arithmetic-geometric-harmonic means inequality This is version 4 of harmonic mean, born on 2001-10-20, modified 2002-02-21. I think that the name "harmonic series" is due to this fact. planetmath.org /encyclopedia/HarmonicMean.html   (175 words)

 Dorlands Medical Dictionary 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 population mean,   the mean of the probability distribution characterizing a specified population; for a finite population, the arithmetic mean of the population values. k[schwa]-pe)  examination of the mediastinum by means of an endoscope inserted through an anterior incision in the suprasternal notch, permitting direct inspection and biopsy of tissue in the anterior superior mediastinum. www.mercksource.com /pp/us/cns/cns_hl_dorlands.jspzQzpgzEzzSzppdocszSzuszSzcommonzSzdorlandszSzdorlandzSzdmd_m_06zPzhtm   (5026 words)

 Iterated Means   (Site not responding. Last check: 2007-10-19) The three most common "means" are a + c Arithmetic mean: A(a,c) = ------- 2 Geometric mean: G(a,c) = sqrt(ac) 2 Harmonic mean: H(a,c) = ------------- (1/a) + (1/c) In their usual thorough fashion the ancient Greeks articulated TEN distinct "means", including the three above. The case c/a gives the "subcontrary to the harmonic mean" a^2 + c^2 K(a,c) = ----------- a + c whereas the cases b/a and c/b give the peculiar "means" _______________________ / a-c \ Since the AG and GH iterated means are so interesting, it seems that it might be worthwhile to investigate the other iterated means. www.mathpages.com /home/kmath130.htm   (486 words)

 Encyclopedia: Weighted harmonic mean 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. In statistics, given a set of data, X = { x1, x2,..., xn} and corresponding weights, W = { w1, w2,..., wn} the weighted geometric mean is calculated as Note that if all the weights are equal, the weighted geometric mean is the same as the geometric mean. In statistics, central tendency is an average of a set of measurements, the word average being variously construed as mean, median, or other measure of location, depending on the context. www.nationmaster.com /encyclopedia/Weighted-harmonic-mean   (374 words)

 MSN Encarta - Search Results - harmonic mean Harmonics, series of subsidiary vibrations that accompany a primary, or fundamental, wave-motion vibration, most notably in musical instruments. Mean (mathematics), also known as the arithmetic mean, a value that helps summarize an entire set of numbers. The movement from one chord to another, called a harmonic progression, creates much of the sense of motion in tonal music. encarta.msn.com /harmonic+mean.html   (135 words)

 Lecture 1   (Site not responding. Last check: 2007-10-19) One application of the harmonic mean is to compute an average sample size when the number of subjects used in each cell of an experimental design is different (i.e., when the design is unbalanced). In general the harmonic mean is used when averaging ratio values, as in the problem of determining average trip speed when you travel 30 miles an hour for the first half and 90 miles an hour for the second. Unlike the mean and standard deviation, which are always expressed in their original metrics, skew (and kurtosis) are expressed as pure, dimension-free numbers. www.olemiss.edu /courses/psy501/Lectures/Lecture1/HTML_Files/Lecture1.htm   (1727 words)

 The Harmonic Mean Many courses offered by the School of Computer Science and Engineering use a harmonic mean formula to calculate students' final results given their exam and assignment marks. Such a harmonic mean formula is a way of ensuring that in order to pass the course, students must have performed reasonably well in both aspects of the course. Notice that the arithmetic mean is in all cases 50. www.cse.unsw.edu.au /~teachadmin/info/harmonic3.html   (426 words)

 Harmonic Mean   (Site not responding. Last check: 2007-10-19) If three numbers are such that by whatever part of itself the largest term exceeds the middle term, and the middle term exceeds the third by the same part of the third then the middle term is the harmonic mean of the first and third. Thus the length of the parallel line segment through the intersection of the diagonals is the harmonic mean of the bases of the trapezoid. The harmonic mean is also used to find the average rate. jwilson.coe.uga.edu /emt725/HM/HM.html   (178 words)

 Other descriptive statistics This is obtained by "weighting" the means by 30 and 50, respectively, and dividing by the sum of the wieghts. The harmonic mean is the reciprocal of the arithmetic mean of the reciprocals. Thus the harmonic mean of 2, 5, and 8 is: www.cns.uni.edu /~campbell/stat/cba3.html   (677 words)

 Mean (3 of 4) Annual rate of return for stock portfolio geometric mean The expression: EXP[1.024] means that 2.718 is raised to the 1.024th power. The geometric mean is less affected by extreme values than is the arithmetic mean and is useful as a measure of central tendency for some positively skewed distributions. The geometric mean is an appropriate measure to use for averaging rates. davidmlane.com /hyperstat/A33018.html   (258 words)

 [No title]   (Site not responding. Last check: 2007-10-19) //Harmonic mean program //11/16/2000 /* Write a program that repeatedly asks you to enter pairs of numbers until at least one of the pair is zero. For each pair, the program should use a function to calculate the harmonic mean of the numbers. The harmonic mean of the numbers is the inverse of the average of the inverses and can be calculated as follows: harmonic mean = 2.0 * x * y / (x + y) */ #include dimacs.rutgers.edu /~rkrane/clecs/apa/hmean.cpp   (90 words)

 Geometer's Angle no. 5 by Marcus the Marinite for the Nexus Network Journal vol.3 no.4 Autumn 2001 The outer two numbers are called extremes.[2] A mean is a kind of average between extremes, but the various types of means include averages that are more complex than the familiar concept of 'halfway in between' (i.e., the arithmetic mean). However, if desired, the idea of the mean can be extended beyond its original sense of a particular type of middle place or term to comprise a progression for the purposes of enlarging or reducing elements in an orderly mathematical fashion, and especially for purposes of visual perspective, the diminution of forms in pictorial space. Applying a mean to determine a progression is quite simple in the case of the arithmetic and geometric means, but in the case of the harmonic mean, it is not nearly as straightforward. www.mat.ub.es /EMIS/journals/NNJ/GA-v3n4.html   (1190 words)

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