Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Harmonic mean

From Wikipedia, the free encyclopedia
Inverse of the average of the inverses of a set of numbers

Inmathematics, theharmonic mean is a kind ofaverage, one of thePythagorean means.

It is the most appropriate average forratios andrates such as speeds,[1][2] and is normally only used for positive arguments.[3]

The harmonic mean is thereciprocal of thearithmetic mean of the reciprocals of the numbers, that is, thegeneralized f-mean withf(x)=1x{\displaystyle f(x)={\frac {1}{x}}}. For example, the harmonic mean of 1, 4, and 4 is

(1−1+4−1+4−13)−1=311+14+14=31.5=2.{\displaystyle \left({\frac {1^{-1}+4^{-1}+4^{-1}}{3}}\right)^{-1}={\frac {3}{{\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{4}}}}={\frac {3}{1.5}}=2\,.}

Definition

[edit]

The harmonic meanH of the positivereal numbersx1,x2,…,xn{\displaystyle x_{1},x_{2},\ldots ,x_{n}} is[4]

H(x1,x2,…,xn)=n1x1+1x2+⋯+1xn=n∑i=1n1xi.{\displaystyle H(x_{1},x_{2},\ldots ,x_{n})={\frac {n}{\displaystyle {\frac {1}{x_{1}}}+{\frac {1}{x_{2}}}+\cdots +{\frac {1}{x_{n}}}}}={\frac {n}{\displaystyle \sum _{i=1}^{n}{\frac {1}{x_{i}}}}}.}

It is the reciprocal of thearithmetic mean of the reciprocals, and vice versa:

H(x1,x2,…,xn)=1A(1x1,1x2,…1xn),A(x1,x2,…,xn)=1H(1x1,1x2,…1xn),{\displaystyle {\begin{aligned}H(x_{1},x_{2},\ldots ,x_{n})&={\frac {1}{\displaystyle A\left({\frac {1}{x_{1}}},{\frac {1}{x_{2}}},\ldots {\frac {1}{x_{n}}}\right)}},\\A(x_{1},x_{2},\ldots ,x_{n})&={\frac {1}{\displaystyle H\left({\frac {1}{x_{1}}},{\frac {1}{x_{2}}},\ldots {\frac {1}{x_{n}}}\right)}},\end{aligned}}}

where the arithmetic mean isA(x1,x2,…,xn)=1n∑i=1nxi.{\textstyle A(x_{1},x_{2},\ldots ,x_{n})={\tfrac {1}{n}}\sum _{i=1}^{n}x_{i}.}

The harmonic mean is aSchur-concave function, and is greater than or equal to the minimum of its arguments: for positive arguments,min(x1…xn)≤H(x1…xn)≤nmin(x1…xn){\displaystyle \min(x_{1}\ldots x_{n})\leq H(x_{1}\ldots x_{n})\leq n\min(x_{1}\ldots x_{n})}. Thus, the harmonic mean cannot be madearbitrarily large by changing some values to bigger ones (while having at least one value unchanged).[citation needed]

The harmonic mean is alsoconcave for positive arguments, an even stronger property than Schur-concavity.[citation needed]

Relationship with other means

[edit]
Geometricproof without words thatmax (a,b) >root mean square (RMS) orquadratic mean (QM) >arithmetic mean (AM) >geometric mean (GM) >harmonic mean (HM) >min (a,b) of two distinct positive numbersa andb[note 1]

For allpositive data setscontaining at least one pair of nonequal values, the harmonic mean is always the least of the three Pythagorean means,[5] while thearithmetic mean is always the greatest of the three and thegeometric mean is always in between. (If all values in a nonempty data set are equal, the three means are always equal.)

It is the special caseM−1 of thepower mean:H(x1,x2,…,xn)=M−1(x1,x2,…,xn)=nx1−1+x2−1+⋯+xn−1{\displaystyle H\left(x_{1},x_{2},\ldots ,x_{n}\right)=M_{-1}\left(x_{1},x_{2},\ldots ,x_{n}\right)={\frac {n}{x_{1}^{-1}+x_{2}^{-1}+\cdots +x_{n}^{-1}}}}

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 mistakenly used in places calling for the harmonic mean.[6] In the speed examplebelow for instance, the arithmetic mean of 40 is incorrect, and too big.

The harmonic mean is related to the other Pythagorean means, as seen in the equation below. This can be seen by interpreting the denominator to be the arithmetic mean of the product of numbersn times but each time omitting thej-th term. That is, for the first term, we multiply alln numbers except the first; for the second, we multiply alln numbers except the second; and so on. The numerator, excluding then, which goes with the arithmetic mean, is the geometric mean to the power n. Thus then-th harmonic mean is related to then-th geometric and arithmetic means. The general formula isH(x1,…,xn)=(G(x1,…,xn))nA(x2x3⋯xn,x1x3⋯xn,…,x1x2⋯xn−1)=(G(x1,…,xn))nA(1x1∏i=1nxi,1x2∏i=1nxi,…,1xn∏i=1nxi).{\displaystyle H\left(x_{1},\ldots ,x_{n}\right)={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left(x_{2}x_{3}\cdots x_{n},x_{1}x_{3}\cdots x_{n},\ldots ,x_{1}x_{2}\cdots x_{n-1}\right)}}={\frac {\left(G\left(x_{1},\ldots ,x_{n}\right)\right)^{n}}{A\left({\frac {1}{x_{1}}}{\prod \limits _{i=1}^{n}x_{i}},{\frac {1}{x_{2}}}{\prod \limits _{i=1}^{n}x_{i}},\ldots ,{\frac {1}{x_{n}}}{\prod \limits _{i=1}^{n}x_{i}}\right)}}.}

If a set of non-identical numbers is subjected to amean-preserving spread — that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.[7]

Harmonic mean of two or three numbers

[edit]

Two numbers

[edit]
A geometric construction of the threePythagorean means of two numbers,a andb. The harmonic mean is denoted byH in purple, while thearithmetic mean isA in red and thegeometric mean isG in blue.Q denotes a fourth mean, thequadratic mean. Since ahypotenuse is always longer than a leg of aright triangle, the diagram shows thatH≤G≤A≤Q{\displaystyle H\leq G\leq A\leq Q}.
A graphical interpretation of the harmonic mean,z of two numbers,x andy, and anomogram to calculate it. The blue line shows that the harmonic mean of 6 and 2 is 3. The magenta line shows that the harmonic mean of 6 and −2 is −6. The red line shows that the harmonic mean of a number and its negative is undefined as the line does not intersect thez axis.

For the special case of just two numbers,x1{\displaystyle x_{1}} andx2{\displaystyle x_{2}}, the harmonic mean can be written as:[4]

H=2x1x2x1+x2{\displaystyle H={\frac {2x_{1}x_{2}}{x_{1}+x_{2}}}\qquad } or1H=(1/x1)+(1/x2)2.{\displaystyle \qquad {\frac {1}{H}}={\frac {(1/x_{1})+(1/x_{2})}{2}}.}

(Note that the harmonic mean is undefined ifx1+x2=0{\displaystyle x_{1}+x_{2}=0}, i.e.x1=−x2{\displaystyle x_{1}=-x_{2}}.)

In this special case, the harmonic mean is related to thearithmetic meanA=x1+x22{\displaystyle A={\frac {x_{1}+x_{2}}{2}}} and thegeometric meanG=x1x2,{\displaystyle G={\sqrt {x_{1}x_{2}}},} by[4]

H=G2A=G(GA).{\displaystyle H={\frac {G^{2}}{A}}=G\left({\frac {G}{A}}\right).}

SinceGA≤1{\displaystyle {\tfrac {G}{A}}\leq 1} by theinequality of arithmetic and geometric means, this shows for then = 2 case thatH ≤G (a property that in fact holds for alln). It also follows thatG=AH{\displaystyle G={\sqrt {AH}}}, meaning the two numbers' geometric mean equals the geometric mean of their arithmetic and harmonic means.

Three numbers

[edit]

For the special case of three numbers,x1{\displaystyle x_{1}},x2{\displaystyle x_{2}} andx3{\displaystyle x_{3}}, the harmonic mean can be written as:[4]

H=3x1x2x3x1x2+x1x3+x2x3.{\displaystyle H={\frac {3x_{1}x_{2}x_{3}}{x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}}}.}

Three positive numbersH,G, andA are respectively the harmonic, geometric, and arithmetic means of three positive numbersif and only if[8]: p.74, #1834  the following inequality holds

A3G3+G3H3+1≤34(1+AH)2.{\displaystyle {\frac {A^{3}}{G^{3}}}+{\frac {G^{3}}{H^{3}}}+1\leq {\frac {3}{4}}\left(1+{\frac {A}{H}}\right)^{2}.}

Weighted harmonic mean

[edit]

If a set ofweightsw1{\displaystyle w_{1}}, ...,wn{\displaystyle w_{n}} is associated to the data setx1{\displaystyle x_{1}}, ...,xn{\displaystyle x_{n}}, theweighted harmonic mean is defined by[9]

H=∑i=1nwi∑i=1nwixi=(∑i=1nwixi−1∑i=1nwi)−1.{\displaystyle H={\frac {\sum \limits _{i=1}^{n}w_{i}}{\sum \limits _{i=1}^{n}{\frac {w_{i}}{x_{i}}}}}=\left({\frac {\sum \limits _{i=1}^{n}w_{i}x_{i}^{-1}}{\sum \limits _{i=1}^{n}w_{i}}}\right)^{-1}.}

The unweighted harmonic mean can be regarded as the special case where all of the weights are equal.

Examples

[edit]

In analytic number theory

[edit]

Prime number theory

[edit]

Theprime number theorem states that the number ofprimes less than or equal ton{\displaystyle n} isasymptotically equal to the harmonic mean of the firstn{\displaystyle n}natural numbers.[10]

In physics

[edit]

Average speed

[edit]

In many situations involvingrates andratios, the harmonic mean provides the correctaverage. For instance, if a vehicle travels a certain distanced outbound at a speedx (e.g. 60 km/h) and returns the same distance at a speedy (e.g. 20 km/h), then its average speed is the harmonic mean ofx andy (30 km/h), not the arithmetic mean (40 km/h). The total travel time is the same as if it had traveled the whole distance at that average speed. This can be proven as follows:[11]

Average speed for the entire journey=⁠Total distance traveled/Sum of time for each segment⁠=⁠2d/⁠d/x⁠ +⁠d/y⁠⁠ =⁠2xy/x +y⁠

However, if the vehicle travels for a certain amount oftime at a speedx and then the same amount of time at a speedy, then its average speed is thearithmetic mean ofx andy, which in the above example is 40 km/h.

Average speed for the entire journey=⁠Total distance traveled/Sum of time for each segment⁠=⁠xt +yt/2t⁠=⁠x +y/2⁠

The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the samedistance, then the average speed is theharmonic mean of all the sub-trip speeds; and if each sub-trip takes the same amount oftime, then the average speed is thearithmetic mean of all the sub-trip speeds. (If neither is the case, then aweighted harmonic mean orweighted arithmetic mean is needed. For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding weight is the distance. In both cases, the resulting formula reduces to dividing the total distance by the total time.)

However, one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the inverse of speed. When trip slowness is found, invert it so as to find the "true" average trip speed. For each trip segment i, the slowness si = 1/speedi. Then take the weightedarithmetic mean of the si's weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average slowness (in time per kilometre). It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean. Thus it illustrates why the harmonic mean works in this case.

Density

[edit]
This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(December 2019) (Learn how and when to remove this message)

Similarly, if one wishes to estimate the density of analloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applyingdimensional analysis to the problem while labeling the mass units by element and making sure that only like element-masses cancel makes this clear.

Electricity

[edit]
See also:Parallel (operator)

If one connects two electricalresistors in parallel, one having resistancex (e.g., 60 Ω) and one having resistancey (e.g., 40 Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean ofx andy (48 Ω): the equivalent resistance, in either case, is 24 Ω (one-half of the harmonic mean). This same principle applies tocapacitors in series or toinductors in parallel.

However, if one connects the resistors in series, then the average resistance is the arithmetic mean ofx andy (50 Ω), with total resistance equal to twice this, the sum ofx andy (100 Ω). This principle applies tocapacitors in parallel or toinductors in series.

As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series.

The "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along the three crystallographic directions.[12]

Optics

[edit]

As for otheroptic equations, thethin lens equation⁠1/f⁠ =⁠1/u⁠ +⁠1/v⁠ can be rewritten such that the focal lengthf is one-half of the harmonic mean of the distances of the subjectu and objectv from the lens.[13]

Two thin lenses of focal lengthf1 andf2 in series is equivalent to two thin lenses of focal lengthfhm, their harmonic mean, in series. Expressed asoptical power, two thin lenses of optical powersP1 andP2 in series is equivalent to two thin lenses of optical powerPam, their arithmetic mean, in series.

In finance

[edit]

The weighted harmonic mean is the preferable method for averaging multiples, such as theprice–earnings ratio (P/E). If these ratios are averaged using a weighted arithmetic mean, high data points are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point.[14] The simple weighted arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey (see above).[15]

In geometry

[edit]

In anytriangle, the radius of theincircle is one-third of the harmonic mean of thealtitudes.

For any point P on theminor arc BC of thecircumcircle of anequilateral triangle ABC, with distancesq andt from B and C respectively, and with the intersection of PA and BC being at a distancey from point P, we have thaty is half the harmonic mean ofq andt.[16]

In aright triangle with legsa andb andaltitudeh from thehypotenuse to the right angle,h2 is half the harmonic mean ofa2 andb2.[17][18]

Lett ands (t >s) be the sides of the twoinscribed squares in a right triangle with hypotenusec. Thens2 equals half the harmonic mean ofc2 andt2.

Let atrapezoid have vertices A, B, C, and D in sequence and have parallel sides AB and CD. Let E be the intersection of thediagonals, and let F be on side DA and G be on side BC such that FEG is parallel to AB and CD. Then FG is the harmonic mean of AB and DC. (This is provable using similar triangles.)

Crossed ladders.h is half the harmonic mean ofA andB

One application of this trapezoid result is in thecrossed ladders problem, where two ladders lie oppositely across an alley, each with feet at the base of one sidewall, with one leaning against a wall at heightA and the other leaning against the opposite wall at heightB, as shown. The ladders cross at a height ofh above the alley floor. Thenh is half the harmonic mean ofA andB. This result still holds if the walls are slanted but still parallel and the "heights"A,B, andh are measured as distances from the floor along lines parallel to the walls. This can be proved easily using the area formula of a trapezoid and area addition formula.

In anellipse, thesemi-latus rectum (the distance from a focus to the ellipse along a line parallel to the minor axis) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus.

In other sciences

[edit]

Incomputer science, specificallyinformation retrieval andmachine learning, the harmonic mean of theprecision (true positives per predicted positive) and therecall (true positives per real positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: theF-score (or F-measure). This is used in information retrieval because only the positive class is ofrelevance, while number of negatives, in general, is large and unknown.[19] It is thus a trade-off as to whether the correct positive predictions should be measured in relation to the number of predicted positives or the number of real positives, so it is measured versus a putative number of positives that is an arithmetic mean of the two possible denominators.

A consequence arises from basic algebra in problems where people or systems work 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 pumps⁠6·4/6 + 4⁠, which is equal to 2.4 hours, to drain the pool together. This is one-half of the harmonic mean of 6 and 4:⁠2·6·4/6 + 4⁠ = 4.8. That is, the appropriate average for the two types of pump is the harmonic mean, and with one pair of pumps (two pumps), it takes half this harmonic mean time, while with two pairs of pumps (four pumps) it would take a quarter of this harmonic mean time.

Inhydrology, the harmonic mean is similarly used to averagehydraulic conductivity values for a flow that is perpendicular to layers (e.g., geologic or soil) - flow parallel to layers uses the arithmetic mean. This apparent difference in averaging is explained by the fact that hydrology uses conductivity, which is the inverse of resistivity.

Insabermetrics, a baseball player'sPower–speed number is the harmonic mean of theirhome run andstolen base totals.

Inpopulation genetics, the harmonic mean is used when calculating the effects of fluctuations in the census population size on the effective population size. The harmonic mean takes into account the fact that events such as populationbottleneck increase the rate genetic drift and reduce the amount of genetic variation in the population. This is a result of the fact that following a bottleneck very few individuals contribute to thegene pool limiting the genetic variation present in the population for many generations to come.

When consideringfuel economy in automobiles two measures are commonly used – miles per gallon (mpg), and litres per 100 km. As the dimensions of these quantities are the inverse of each other (one is distance per volume, the other volume per distance) when taking the mean value of the fuel economy of a range of cars one measure will produce the harmonic mean of the other – i.e., converting the mean value of fuel economy expressed in litres per 100 km to miles per gallon will produce the harmonic mean of the fuel economy expressed in miles per gallon. For calculating the average fuel consumption of a fleet of vehicles from the individual fuel consumptions, the harmonic mean should be used if the fleet uses miles per gallon, whereas the arithmetic mean should be used if the fleet uses litres per 100 km. In the USA theCAFE standards (the federal automobile fuel consumption standards) make use of the harmonic mean.

Inchemistry andnuclear physics the average mass per particle of a mixture consisting of different species (e.g., molecules or isotopes) is given by the harmonic mean of the individual species' masses weighted by their respective mass fraction.

Beta distribution

[edit]
This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(December 2019) (Learn how and when to remove this message)
Harmonic mean for Beta distribution for 0 < α < 5 and 0 < β < 5
(Mean - HarmonicMean) for Beta distribution versus alpha and beta from 0 to 2
Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), smaller values alpha and beta in front
Harmonic Means for Beta distribution Purple=H(X), Yellow=H(1-X), larger values alpha and beta in front

The harmonic mean of abeta distribution with shape parametersα andβ is:

H=α−1α+β−1 conditional on α>1&β>0{\displaystyle H={\frac {\alpha -1}{\alpha +\beta -1}}{\text{ conditional on }}\alpha >1\,\,\&\,\,\beta >0}

The harmonic mean withα < 1 is undefined because its defining expression is not bounded in [0, 1].

Lettingα =β

H=α−12α−1{\displaystyle H={\frac {\alpha -1}{2\alpha -1}}}

showing that forα =β the harmonic mean ranges from 0 forα =β = 1, to 1/2 forα =β → ∞.

The following are the limits with one parameter finite (non-zero) and the other parameter approaching these limits:

limα→0H= undefined limα→1H=limβ→∞H=0limβ→0H=limα→∞H=1{\displaystyle {\begin{aligned}\lim _{\alpha \to 0}H&={\text{ undefined }}\\\lim _{\alpha \to 1}H&=\lim _{\beta \to \infty }H=0\\\lim _{\beta \to 0}H&=\lim _{\alpha \to \infty }H=1\end{aligned}}}

With the geometric mean the harmonic mean may be useful in maximum likelihood estimation in the four parameter case.

A second harmonic mean (H1 − X) also exists for this distribution

H1−X=β−1α+β−1 conditional on β>1&α>0{\displaystyle H_{1-X}={\frac {\beta -1}{\alpha +\beta -1}}{\text{ conditional on }}\beta >1\,\,\&\,\,\alpha >0}

This harmonic mean withβ < 1 is undefined because its defining expression is not bounded in [ 0, 1 ].

Lettingα =β in the above expression

H1−X=β−12β−1{\displaystyle H_{1-X}={\frac {\beta -1}{2\beta -1}}}

showing that forα =β the harmonic mean ranges from 0, forα =β = 1, to 1/2, forα =β → ∞.

The following are the limits with one parameter finite (non zero) and the other approaching these limits:

limβ→0H1−X= undefined limβ→1H1−X=limα→∞H1−X=0limα→0H1−X=limβ→∞H1−X=1{\displaystyle {\begin{aligned}\lim _{\beta \to 0}H_{1-X}&={\text{ undefined }}\\\lim _{\beta \to 1}H_{1-X}&=\lim _{\alpha \to \infty }H_{1-X}=0\\\lim _{\alpha \to 0}H_{1-X}&=\lim _{\beta \to \infty }H_{1-X}=1\end{aligned}}}

Although both harmonic means are asymmetric, whenα =β the two means are equal.

Lognormal distribution

[edit]

The harmonic mean (H ) of thelognormal distribution of a random variableX is[20]

H=exp⁡(μ−12σ2),{\displaystyle H=\exp \left(\mu -{\frac {1}{2}}\sigma ^{2}\right),}

whereμ andσ2 are the parameters of the distribution, i.e. the mean and variance of the distribution of the natural logarithm ofX.

The harmonic and arithmetic means of the distribution are related by

μ∗H=1+Cv2,{\displaystyle {\frac {\mu ^{*}}{H}}=1+C_{v}^{2}\,,}

whereCv andμ* are thecoefficient of variation and the mean of the distribution respectively..

The geometric (G), arithmetic and harmonic means of the distribution are related by[21]

Hμ∗=G2.{\displaystyle H\mu ^{*}=G^{2}.}

Pareto distribution

[edit]

The harmonic mean of type 1Pareto distribution is[22]

H=k(1+1α){\displaystyle H=k\left(1+{\frac {1}{\alpha }}\right)}

wherek is the scale parameter andα is the shape parameter.

Statistics

[edit]

For a random sample, the harmonic mean is calculated as above. Both themean and thevariance may beinfinite (if it includes at least one term of the form 1/0).

Sample distributions of mean and variance

[edit]

The mean of the samplem is asymptotically distributed normally with variances2.

s2=m[E⁡(1x−1)]m2n{\displaystyle s^{2}={\frac {m\left[\operatorname {E} \left({\frac {1}{x}}-1\right)\right]}{m^{2}n}}}

The variance of the mean itself is[23]

Var⁡(1x)=m[E⁡(1x−1)]nm2{\displaystyle \operatorname {Var} \left({\frac {1}{x}}\right)={\frac {m\left[\operatorname {E} \left({\frac {1}{x}}-1\right)\right]}{nm^{2}}}}

wherem is the arithmetic mean of the reciprocals,x are the variates,n is the population size andE is the expectation operator.

Delta method

[edit]
This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.(December 2019) (Learn how and when to remove this message)

Assuming that the variance is not infinite and that thecentral limit theorem applies to the sample then using thedelta method, the variance is

Var⁡(H)=1ns2m4{\displaystyle \operatorname {Var} (H)={\frac {1}{n}}{\frac {s^{2}}{m^{4}}}}

whereH is the harmonic mean,m is the arithmetic mean of the reciprocals

m=1n∑1x.{\displaystyle m={\frac {1}{n}}\sum {\frac {1}{x}}.}

s2 is the variance of the reciprocals of the data

s2=Var⁡(1x){\displaystyle s^{2}=\operatorname {Var} \left({\frac {1}{x}}\right)}

andn is the number of data points in the sample.

Jackknife method

[edit]

Ajackknife method of estimating the variance is possible if the mean is known.[24] This method is the usual 'delete 1' rather than the 'delete m' version.

This method first requires the computation of the mean of the sample (m)

m=n∑1x{\displaystyle m={\frac {n}{\sum {\frac {1}{x}}}}}

wherex are the sample values.

A series of valuewi is then computed where

wi=n−1∑j≠i1x.{\displaystyle w_{i}={\frac {n-1}{\sum _{j\neq i}{\frac {1}{x}}}}.}

The mean (h) of thewi is then taken:

h=1n∑wi{\displaystyle h={\frac {1}{n}}\sum {w_{i}}}

The variance of the mean is

n−1n∑(m−wi)2.{\displaystyle {\frac {n-1}{n}}\sum {(m-w_{i})}^{2}.}

Significance testing andconfidence intervals for the mean can then be estimated with thet test.

Size biased sampling

[edit]

Assume a random variate has a distributionf(x ). Assume also that the likelihood of a variate being chosen is proportional to its value. This is known as length based or size biased sampling.

Letμ be the mean of the population. Then theprobability density functionf*(x ) of the size biased population is

f∗(x)=xf(x)μ{\displaystyle f^{*}(x)={\frac {xf(x)}{\mu }}}

The expectation of this length biased distribution E*(x ) is[23]

E∗⁡(x)=μ[1+σ2μ2]{\displaystyle \operatorname {E} ^{*}(x)=\mu \left[1+{\frac {\sigma ^{2}}{\mu ^{2}}}\right]}

whereσ2 is the variance.

The expectation of the harmonic mean is the same as the non-length biased version E(x )

E∗(x−1)=E(x)−1{\displaystyle E^{*}(x^{-1})=E(x)^{-1}}

The problem of length biased sampling arises in a number of areas including textile manufacture[25] pedigree analysis[26] and survival analysis[27]

Akmanet al. have developed a test for the detection of length based bias in samples.[28]

Shifted variables

[edit]

IfX is a positive random variable andq > 0 then for allε > 0[29]

Var⁡[1(X+ϵ)q]<Var⁡(1Xq).{\displaystyle \operatorname {Var} \left[{\frac {1}{(X+\epsilon )^{q}}}\right]<\operatorname {Var} \left({\frac {1}{X^{q}}}\right).}

Moments

[edit]

Assuming thatX and E(X) are > 0 then[29]

E⁡[1X]≥1E⁡(X){\displaystyle \operatorname {E} \left[{\frac {1}{X}}\right]\geq {\frac {1}{\operatorname {E} (X)}}}

This follows fromJensen's inequality.

Gurland has shown that[30] for a distribution that takes only positive values, for anyn > 0

E⁡(X−1)≥E⁡(Xn−1)E⁡(Xn).{\displaystyle \operatorname {E} \left(X^{-1}\right)\geq {\frac {\operatorname {E} \left(X^{n-1}\right)}{\operatorname {E} \left(X^{n}\right)}}.}

Under some conditions[31]

E⁡(a+X)−n∼E⁡(a+X−n){\displaystyle \operatorname {E} (a+X)^{-n}\sim \operatorname {E} \left(a+X^{-n}\right)}

where ~ means approximately equal to.

Sampling properties

[edit]

Assuming that the variates (x) are drawn from a lognormal distribution there are several possible estimators forH:

H1=n∑(1x)H2=(exp⁡[1n∑loge⁡(x)])21n∑(x)H3=exp⁡(m−12s2){\displaystyle {\begin{aligned}H_{1}&={\frac {n}{\sum \left({\frac {1}{x}}\right)}}\\H_{2}&={\frac {\left(\exp \left[{\frac {1}{n}}\sum \log _{e}(x)\right]\right)^{2}}{{\frac {1}{n}}\sum (x)}}\\H_{3}&=\exp \left(m-{\frac {1}{2}}s^{2}\right)\end{aligned}}}

where

m=1n∑loge⁡(x){\displaystyle m={\frac {1}{n}}\sum \log _{e}(x)}
s2=1n∑(loge⁡(x)−m)2{\displaystyle s^{2}={\frac {1}{n}}\sum \left(\log _{e}(x)-m\right)^{2}}

Of theseH3 is probably the best estimator for samples of 25 or more.[32]

Bias and variance estimators

[edit]

A first order approximation to thebias and variance ofH1 are[33]

bias⁡[H1]=HCvnVar⁡[H1]=H2Cvn{\displaystyle {\begin{aligned}\operatorname {bias} \left[H_{1}\right]&={\frac {HC_{v}}{n}}\\\operatorname {Var} \left[H_{1}\right]&={\frac {H^{2}C_{v}}{n}}\end{aligned}}}

whereCv is the coefficient of variation.

Similarly a first order approximation to the bias and variance ofH3 are[33]

Hloge⁡(1+Cv)2n[1+1+Cv22]Hloge⁡(1+Cv)n[1+1+Cv24]{\displaystyle {\begin{aligned}{\frac {H\log _{e}\left(1+C_{v}\right)}{2n}}\left[1+{\frac {1+C_{v}^{2}}{2}}\right]\\{\frac {H\log _{e}\left(1+C_{v}\right)}{n}}\left[1+{\frac {1+C_{v}^{2}}{4}}\right]\end{aligned}}}

In numerical experimentsH3 is generally a superior estimator of the harmonic mean thanH1.[33]H2 produces estimates that are largely similar toH1.

Notes

[edit]

TheEnvironmental Protection Agency recommends the use of the harmonic mean in setting maximum toxin levels in water.[34]

In geophysicalreservoir engineering studies, the harmonic mean is widely used.[35]

See also

[edit]

Notes

[edit]
  1. ^If NM =a and PM =b. AM =AM ofa andb, and radiusr = AQ = AG.
    UsingPythagoras' theorem, QM² = AQ² + AM² ∴ QM = √AQ² + AM² =QM.
    Using Pythagoras' theorem, AM² = AG² + GM² ∴ GM = √AM² − AG² =GM.
    Usingsimilar triangles,⁠HM/GM⁠ =⁠GM/AM⁠ ∴ HM =⁠GM²/AM⁠ =HM.

References

[edit]
  1. ^CourseArchived 2022-07-11 at theWayback Machine
  2. ^Srivastava, U. K.; Shenoy, G. V.; Sharma, S. C. (1989).Quantitative Techniques for Managerial Decisions. New Age International. p. 63.ISBN 978-81-224-0189-9.
  3. ^Jones, Alan (2018-10-09).Probability, Statistics and Other Frightening Stuff. Routledge. p. 42.ISBN 978-1-351-66138-6.
  4. ^abcdWeisstein, Eric W."Harmonic Mean".mathworld.wolfram.com. Retrieved2023-05-31.
  5. ^Da-Feng Xia, Sen-Lin Xu, and Feng Qi, "A proof of the arithmetic mean-geometric mean-harmonic mean inequalities", RGMIA Research Report Collection, vol. 2, no. 1, 1999,http://ajmaa.org/RGMIA/papers/v2n1/v2n1-10.pdfArchived 2015-12-22 at theWayback Machine
  6. ^*Statistical Analysis, Ya-lun Chou, Holt International, 1969,ISBN 0030730953
  7. ^Mitchell, Douglas W., "More on spreads and non-arithmetic means,"The Mathematical Gazette 88, March 2004, 142–144.
  8. ^Inequalities proposed in "Crux Mathematicorum","Archived copy"(PDF).Archived(PDF) from the original on 2014-10-15. Retrieved2014-09-09.{{cite web}}: CS1 maint: archived copy as title (link).
  9. ^Ferger F (1931) The nature and use of the harmonic mean. Journal of theAmerican Statistical Association 26(173) 36-40
  10. ^Deveci, Sinan (2022). "On a Double Series Representation of the Natural Logarithm, the Asymptotic Behavior of Hölder Means, and an Elementary Estimate for the Prime Counting Function". p. 2.arXiv:2211.10751 [math.NT].
  11. ^"Average: How to calculate Average, Formula, Weighted average".learningpundits.com.Archived from the original on 29 December 2017. Retrieved8 May 2018.
  12. ^"Effective mass in semiconductors".ecee.colorado.edu. Archived fromthe original on 20 October 2017. Retrieved8 May 2018.
  13. ^Hecht, Eugene (2002).Optics (4th ed.).Addison Wesley. p. 168.ISBN 978-0805385663.
  14. ^"Fairness Opinions: Common Errors and Omissions".The Handbook of Business Valuation and Intellectual Property Analysis. McGraw Hill. 2004.ISBN 0-07-142967-0.
  15. ^Agrrawal, Pankaj; Borgman, Richard; Clark, John M.; Strong, Robert (2010). "Using the Price-to-Earnings Harmonic Mean to Improve Firm Valuation Estimates".Journal of Financial Education.36 (3–4):98–110.ISSN 0093-3961.JSTOR 41948650.SSRN 2621087.
  16. ^Posamentier, Alfred S.; Salkind, Charles T. (1996).Challenging Problems in Geometry (Second ed.). Dover. p. 172.ISBN 0-486-69154-3.
  17. ^Voles, Roger, "Integer solutions ofa−2+b−2=d−2{\displaystyle a^{-2}+b^{-2}=d^{-2}},"Mathematical Gazette 83, July 1999, 269–271.
  18. ^Richinick, Jennifer, "The upside-down Pythagorean Theorem,"Mathematical Gazette 92, July 2008, 313–;317.
  19. ^Van Rijsbergen, C. J. (1979).Information Retrieval (2nd ed.). Butterworth.Archived from the original on 2005-04-06.
  20. ^Aitchison J, Brown JAC (1969). The lognormal distribution with special reference to its uses in economics. Cambridge University Press, New York
  21. ^Rossman LA (1990) Design stream flows based on harmonic means. J Hydr Eng ASCE 116(7) 946–950
  22. ^Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions Vol 1. Wiley Series in Probability and Statistics.
  23. ^abZelen M (1972) Length-biased sampling and biomedical problems. In: Biometric Society Meeting, Dallas, Texas
  24. ^Lam FC (1985) Estimate of variance for harmonic mean half lives. J Pharm Sci 74(2) 229-231
  25. ^Cox DR (1969) Some sampling problems in technology. In: New developments in survey sampling. U.L. Johnson, H Smith eds. New York: Wiley Interscience
  26. ^Davidov O, Zelen M (2001) Referent sampling, family history and relative risk: the role of length-biased sampling. Biostat 2(2): 173-181doi:10.1093/biostatistics/2.2.173
  27. ^Zelen M, Feinleib M (1969) On the theory of screening for chronic diseases. Biometrika 56: 601-614
  28. ^Akman O, Gamage J, Jannot J, Juliano S, Thurman A, Whitman D (2007) A simple test for detection of length-biased sampling. J Biostats 1 (2) 189-195
  29. ^abChuen-Teck See, Chen J (2008) Convex functions of random variables. J Inequal Pure Appl Math 9 (3) Art 80
  30. ^Gurland J (1967) An inequality satisfied by the expectation of the reciprocal of a random variable. The American Statistician. 21 (2) 24
  31. ^Sung SH (2010) On inverse moments for a class of nonnegative random variables. J Inequal Applicdoi:10.1155/2010/823767
  32. ^Stedinger JR (1980) Fitting lognormal distributions to hydrologic data. Water Resour Res 16(3) 481–490
  33. ^abcLimbrunner JF, Vogel RM, Brown LC (2000) Estimation of harmonic mean of a lognormal variable. J Hydrol Eng 5(1) 59-66"Archived copy"(PDF). Archived fromthe original(PDF) on 2010-06-11. Retrieved2012-09-16.{{cite web}}: CS1 maint: archived copy as title (link)
  34. ^EPA (1991) Technical support document for water quality-based toxics control. EPA/505/2-90-001. Office of Water
  35. ^Muskat M (1937) The flow of homogeneous fluids through porous media. McGraw-Hill, New York

External links

[edit]
Continuous data
Center
Dispersion
Shape
Count data
Summary tables
Dependence
Graphics
Study design
Survey methodology
Controlled experiments
Adaptive designs
Observational studies
Statistical theory
Frequentist inference
Point estimation
Interval estimation
Testing hypotheses
Parametric tests
Specific tests
Goodness of fit
Rank statistics
Bayesian inference
Correlation
Regression analysis
Linear regression
Non-standard predictors
Generalized linear model
Partition of variance
Categorical
Multivariate
Time-series
General
Specific tests
Time domain
Frequency domain
Survival
Survival function
Hazard function
Test
Biostatistics
Engineering statistics
Social statistics
Spatial statistics
Retrieved from "https://en.wikipedia.org/w/index.php?title=Harmonic_mean&oldid=1284162533"
Category:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp