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Topic: Arithmetic geometric mean

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Inequality of arithmetic and geometric means

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	PlanetMath: proof of arithmetic-geometric-harmonic means inequality
		The proof that the geometric mean is at least as large as the harmonic mean is the usual one (see “proof of arithmetic-geometric-harmonic means inequality”).
		Cross-references: harmonic mean, geometric mean, monotone function, arithmetic mean, concave function, arithmetic-geometric-harmonic means inequality, Jensen inequality
		This is version 4 of proof of arithmetic-geometric-harmonic means inequality, born on 2002-06-03, modified 2003-08-04.
	www.planetmath.org /encyclopedia/UsingJensensInequalityToProveTheArithmeticGeometricHarmonicMeansInequality.html (148 words)

	Geometric mean - Psychology Wiki - a Wikia wiki
		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 geometric mean of a data set is always smaller than or equal to the set's arithmetic mean (the two means are equal if and only if all members of the data set are equal).
		It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.50, and the third year it was multiplied by 1.30.
	psychology.wikia.com /wiki/Geometric_mean (610 words)

	CAS: CASNET Discussion Thread: Arithmetic versus Geometric Mean
		There are other sources (Ibbotson 1999 Yearbook) that says the geometric average is OK for calculating the historical average but you have to use the arithmetic average to calculate a forecast since the arithmetic average is actually the estimate of the mean of the distribution that generated the actual historical series.
		In this case the arithmetic mean equals the geometric mean and the accumulated value is 1 plus the geometric mean raised to the 50th power.
		As the number of consecutive periods increases the geometric mean moves asymptotically towards the limit of the means of the resulting lognormals, which is exp(E[ln(1+r(i))]).
	www.casact.org /forum/index.cfm?fa=discuss2 (3545 words)

	Mean - Psychology Wiki - a Wikia wiki
		The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or most likely (mode).
		The geometric mean is an average that is useful for sets of numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean).
		The generalized mean, also known as the power mean or Hölder mean, is an abstraction of the arithmetic, geometric and harmonic means.
	psychology.wikia.com /wiki/Mean (1127 words)

	PlanetMath: arithmetic-geometric mean
		The AGM can be used to numerically evaluate elliptic integrals of the first and second kinds.
		The fact that relatively few iterations are necessarry to obtain a highly accurate result also means that one does not have to worry much about the cumulative effect of roundoff errors in the various steps of the computation.
		This is version 4 of arithmetic-geometric mean, born on 2004-06-05, modified 2007-05-27.
	www.planetmath.org /encyclopedia/ArithmeticGeometricMean.html (304 words)

	Geometric mean arithmetic mean or means calculate center frequeny calculation average numbers bandwidth - sengpielaudio
		The geometric mean of two numbers is the square root of their product.
		The geometric mean of three numbers is the cubic root of their product.
		The arithmetic mean is the sum of the numbers, divided by the quantity of the numbers.
	www.sengpielaudio.com /calculator-geommean.htm (151 words)

	The Super-Symmetric Mean
		The famous Arithmetic-Geometric Mean (AGM) has found many uses in analysis (e.g., elliptic integrals) and number theory (e.g., quadratically convergent algorithms for computing the digits of PI), but there does not seem to have been much attention devoted to higher-order versions of iterated means of more than two arguments.
		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.
		To illustrate on a simple example, Figure 1 shows how the arithmetic and geometric means of the numbers 3 and 15 are related to the two polynomials that match the value and the derivative of the function f(x)=(x-3)(x-15) respectively at x=0.
	www.mathpages.com /home/kmath461.htm (768 words)

	Arithmetic vs. Geometric Mean
		The arithmetic standard deviation is a quantity. Multiples of that quantity are added to (or subtracted from) the arithmetic mean to determine the set of values that lie within a given range of dispersion. To determine the value that lies n standard deviations from the mean, we must multiply the arithmetic standard deviation by n.
		The arithmetic mean and arithmetic standard deviation are sum-based values. As such, they are appropriate for additive processes. But earnings growth is not additive; it is multiplicative. The appropriate measures for EPS growth factors are the geometric mean and geometric standard deviation, which are product-based values.
		The arithmetic mean of these factors is 1.2584, implying that EPS grew at an average annual rate of 25.84%. Had that been so, the first EPS, $0.32, would have grown into $2.53 in the last year—which, of course, it did not.
	www.thinkingapplied.com /means_folder/deceptive_means.htm (1949 words)

	Statistics of central tendency
		Arithmetic mean: The arithmetic mean is the sum of the observations divided by the number of observations.
		The geometric mean is slightly smaller than the arithmetic mean; unless the data are highly skewed, the difference between the arithmetic and geometric means is small.
		The harmonic mean is less sensitive to a few large values than are the arithmetic or geometric mean, so it is sometimes used for variables such as dispersal distance.
	udel.edu /~mcdonald/statcentral.html (1437 words)

	Arithmetic/Geometric Means Essay (Site not responding. Last check: )
		First, we will prove that the arithmetic means is always greater than or equal to the geometric means by an algebraic proof.
		A comparison of arithmetic mean and geometric mean of a set a values for a and b can be easily visualized in a spreadsheet.
		The arithmetic mean is calculated in column C, with the geometric mean stored in column D.
	jwilson.coe.uga.edu /emt669/student.folders/callinan.michael/Essays/Essay2/Essay2.html (450 words)

	Riemann for Anti-Dummies, Part 66
		For example, the visual apparatus cannot perceive the proportionality nor the incommensurability of the geometric mean with respect to the doubling of the square, as acutely as the ears perceive a congruent relationship with respect to the Lydian interval.
		Thus, the arithmetic mean expresses a principle of change in which the change produces effects that are equal, and the differences between such effects are equal--such as in the extension of a line.
		The harmonic mean expresses the inverse of the arithmetic, when the differences between the mean and the extremes are in the same proportion to each other as the extremes are to themselves, as in the case of the hyperbola.
	www.wlym.com /antidummies/part66.html (7743 words)

	How to Calculate Geometric Means (Site not responding. Last check: )
		A geometric mean, unlike an arithmetic mean, tends to dampen the effect of very high or low values, which might bias the mean if a straight average (arithmetic mean) were calculated.
		Geometric mean is often used to evaluate data covering several orders of magnitude, and sometimes for evaluating ratios, percentages, or other data sets bounded by zero.
		For example, to calculate the geometric mean of the values +12%, -8%, and +2%, instead calculate the geometric mean of their decimal multiplier equivalents of 1.12, 0.92, and 1.02, to compute a geometric mean of 1.0167.
	www.buzzardsbay.org /geomean.htm (3107 words)

	Means: Arithmetic, Geometric and Golden
		The term "mean" in mathematics simply reflects a specific relationship of one number as the middle point of two extremes.
		The arithmetic mean is thus the simple average between two numbers.
		The geometric mean is similar, but based on a common multiplier that relates the mean to the other two numbers.
	evolutionoftruth.com /goldensection/means.htm (293 words)

	[No title]
		The mean or arithmetic mean is one measure of the centre of a set of data.
		The mean is calculated by adding up all the values and dividing the total by the total number of values.
		The case t = 1 yields the arithmetic mean and the case t = −1 yields the harmonic mean.
	www.lycos.com /info/arithmetic-mean--numbers.html (261 words)

	Mean
		The mean for a given set of numbers can be computed in more than one way, including the arithmetic mean method, which uses the sum of the numbers in the series, and the geometric mean method.
		If stock XYZ closed at $50, $51 and $54 over the past three days, the arithmetic mean would be the sum of those numbers divided by three, which is $51.67.
		In contrast, the geometric mean would be computed as third root of the numbers' product, or the third root of 137,700, which approximately equals $51.64.
	www.investopedia.com /terms/m/mean.asp (274 words)

	SPECviewperf® 7.1 -- Weighted Geometric Mean
		Above is the formula for determining a weighted geometric mean, where "n" is the number of individual tests in a viewset, and "w" is the weight of each individual test, expressed as a number between 0.0 and 1.0.
		The weighted arithmetic mean is correct for calculating grades at the end of a school term.
		If the weighted geometric mean of System B is 10-percent higher than System A, for example, the normalized weighted geometric mean of System B will be 10-percent higher than System A, no matter what reference system you choose.
	www.spec.org /gpc/opc.static/geometric.html (757 words)

	Definition: Geometric mean
		It is different than the traditional mean (which we sometimes call the arithmetic mean) because it uses multiplication rather than addition to summarize data values.
		The geometric mean is a useful summary when we expect that changes in the data occur in a relative fashion.
		Geometric means are often useful summaries for highly skewed data.
	www.childrensmercy.org /stats/definitions/geometric.htm (506 words)

	[No title]
		When using the arithmetic mean to calculate the mean of a population one makes the assumption that the arithmetic mean is a valid estimate of the central location of a distribution, which is not always true.
		The number needed to treat summarises, by means of a single number, the results of a therapeutic trial in the same way that the arithmetic mean of a variable summarises all the measurements performed on each individual of the sample.
		Nevertheless, the arithmetic mean is only the most probable value of a population (provided the variable is normally distributed) and is not the value observed in every individual.
	www.lycos.com /info/arithmetic-mean.html (359 words)

	Arithmetic-geometric mean - Definition, explanation
		In mathematics, the arithmetic-geometric mean M(x, y) of two positive real numbers x and y is defined as follows: we first form the arithmetic mean of x and y and call it a
		M(x, y) is a number between the geometric and arithmetic mean of x and y; in particular it is between x and y.
		The reciprocal of the arithmetic-geometric mean of 1 and the Square root of 2 is called Gauss's constant.
	www.calsky.com /lexikon/en/txt/a/ar/arithmetic_geometric_mean.php (294 words)

	Arithmetic Mean, Harmonic Mean, Combined Average, Geometric Mean
		The average would simply be the regular arithmetic mean (x1 + x2 + x3)/3 = 1,400 Kchars per 180 seconds or you could convert it to a more meaningful 7.78 Kcharacters/sec.
		What this all means is that if the different samples intuitively should not have the same weights, yet the numerators are all the same, they should be considered as a fixed numerator over a varying denominator.
		The geometric mean is used in cases where the numbers are effectively multipliers rather than having an additive effect.
	home.comcast.net /~b.giese/dc_mean.html (1135 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: )
		Here we cannot use the arithmetic mean and say that the average growth was 3%.
		Note: The mean growth rate comes out to (1.029992 - 1) * 100% = 2.9992% which is very close to the arithmetic mean (as seen in Dr. Floor's answer: compare $109,270,125 to $109,272,700).
		The log of the geometric mean of a set of numbers is the arithmetic mean of the logs of the numbers: log((abc)^(1/3)) = (1/3)log(abc) = (1/3)(log(a)+log(b)+log(c)) Thus you can find the geometric mean by taking the logs of your data, finding the arithmetic mean, then taking the antilog (exponential) of the mean.
	mathforum.org /library/drmath/view/52804.html (646 words)

	MathComplete.com - Sequence and Series - Tutorial (Site not responding. Last check: )
		the arithmetic mean between 5 and 13 is 9, and three arithmetic means between 5 and 21 are 9, 13, 17.
		We are sometimes required to find the arithmetic mean of two numbers X, and Y. This means that we have to insert number i.e.
		the geometric mean between 5 and 20 is 10, and three arithmetic means between 5 and 80 are 10, 20, 40.
	www.mathcomplete.com /tutorial/sequence/?PG=3 (222 words)

	Question Corner -- Applications of the Geometric Mean
		It is not the arithmetic mean, because what these numbers mean is that on the first year your investment was multiplied (not added to) by 1.10, on the second year it was multiplied by 1.60, and the third year it was multiplied by 1.20.
		If you calculate this geometric mean you get approximately 1.283, so the average rate of return is about 28% (not 30% which is what the arithmetic mean of 10%, 60%, and 20% would give you).
		The arithmetic mean can also be interpreted as the length of the sides of a square whose perimeter is the same as our rectangle.
	www.math.toronto.edu /mathnet/questionCorner/geomean.html (2387 words)

	Geometric Mean
		The logarithm of the geometric mean is the arithmetic mean of the log transformed data:
		The geometric mean is an appropriate measure of central tendency when averages of rates or index numbers are required.
		The geometric mean is calculated from the arithmetic mean of the log transformed data.
	shazam.econ.ubc.ca /intro/gmean.htm (195 words)

	Geometric proof: (a+b)÷2 ≥ √(ab)
		This is a fairly random proof of a simple special case of the arithmetic/geometric mean inequality (which is basically that the sum of a set of n positive real numbers, divided by n, is at least as large as the nth root of the product of the same n positive real numbers).
		This means that, in any pair of adjacent triangles (one such pair is shown with thicker lines in Fig.2), the triangles do not overlap at any point.
		As the square has height and width d=2 radii, the semicircles in which opposite triangles are inscribed (highlighted in Fig.3) touch at the centre (i.e.
	www.pseudorandom.co.uk /2000/maths/inequal (341 words)

	[No title]
		This produces a discontinuous manifold of ellipses that converge on a circle whose radius is the arithmetic-geometric mean of the original ellipse.
		This is illustrated by the simplest case: the arithmetic mean between 1 and -1 is 0 which is the mean distance between them and the geometric means are the positive and negative values of the square root of -1, which are the mean angular change between 1 and -1.
		Then (a+c) and (a-c) are the magnitudes whose arithmetic mean is 3, whose geometric mean is 1, and whose arithmetic-geometric mean is 1.86362....
	www.wlym.com /antidummies/part66-old.html (7608 words)

	PHASER: Module: Gauss AGM Pi MAP
		Because of the speed of convergence of the AGM process, as an application, he gives the fastest known algorithm for the computation of π.
		First, he very early noticed that taking the arithmetic and geometric means of two numbers gave two closer numbers, and this process converged quickly to the so called arithmetic-geometric mean (AGM).
		Later he notriced that the AGM of 1 and √2 was the ratio of the arc length of a circle to that of a lemniscate (a closed curve described in polar coordinates by r
	www.phaser.com /modules/historic/gaussagm (255 words)

	avg (Site not responding. Last check: )
		However, ungrouped data are frequently unavailable, and the question arises as to whether an arithmetic or geometric mean is the most appropriate summary measure of exposure.
		It is argued in this paper that one should use the type of mean for which the total risk that would result if every member of the population was exposed to the mean level is as close as possible to the actual total population risk.
		Using this criterion an arithmetic mean is always preferred over a geometric mean whenever the dose response is convex.
	www.angelfire.com /la2/kennycrumpspage/avg.htm (174 words)

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