Movatterモバイル変換

[0]ホーム
URL:

Jump to content
WikipediaThe Free Encyclopedia
Search

Geometric mean

From Wikipedia, the free encyclopedia
N-th root of the product of n numbers
Example of the geometric mean:lg{\displaystyle l_{g}} (red) is the geometric mean ofl1{\displaystyle l_{1}} andl2{\displaystyle l_{2}},[1][2] is an example in which the line segmentl2(BC¯){\displaystyle l_{2}\;({\overline {BC}})} is given as a perpendicular toAB¯{\displaystyle {\overline {AB}}}.AC′¯{\displaystyle {\overline {AC'}}} is the diameter of a circle andBC¯≅BC′¯{\displaystyle {\overline {BC}}\cong {\overline {BC'}}}.

In mathematics, thegeometric mean (also known as themean proportional) is amean oraverage which indicates acentral tendency of a finite collection ofpositive real numbers by using theproduct of their values (as opposed to thearithmetic mean, which uses their sum). The geometric mean of⁠n{\displaystyle n}⁠ numbers is thenth root of their product, i.e., for a collection of numbersa1,a2, ...,an, the geometric mean is defined as

a1a2⋯antn.{\displaystyle {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}.}

When the collection of numbers and their geometric mean are plotted inlogarithmic scale, the geometric mean is transformed into an arithmetic mean, so the geometric mean can equivalently be calculated by taking thenatural logarithm⁠ln{\displaystyle \ln }⁠ of each number, finding the arithmetic mean of the logarithms, and then returning the result to linear scale using theexponential function⁠exp{\displaystyle \exp }⁠,

a1a2⋯antn=exp⁡(ln⁡a1+ln⁡a2+⋯+ln⁡ann).{\displaystyle {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}=\exp \left({\frac {\ln a_{1}+\ln a_{2}+\cdots +\ln a_{n}}{n}}\right).}

The geometric mean of two numbers is thesquare root of their product, for example with numbers⁠2{\displaystyle 2}⁠ and⁠8{\displaystyle 8}⁠ the geometric mean is2⋅8={\displaystyle \textstyle {\sqrt {2\cdot 8}}={}}16=4{\displaystyle \textstyle {\sqrt {16}}=4}. The geometric mean of the three numbers is thecube root of their product, for example with numbers⁠1{\displaystyle 1}⁠,⁠12{\displaystyle 12}⁠, and⁠18{\displaystyle 18}⁠, the geometric mean is1⋅12⋅183={\displaystyle \textstyle {\sqrt[{3}]{1\cdot 12\cdot 18}}={}}2163=6{\displaystyle \textstyle {\sqrt[{3}]{216}}=6}.

The geometric mean is useful whenever the quantities to be averaged combine multiplicatively, such aspopulation growth rates or interest rates of a financial investment. Suppose for example a person invests $1000 and achieves annual returns of +10%, −12%, +90%, −30% and +25%, giving a final value of $1609. The average percentage growth is the geometric mean of the annual growth ratios (1.10, 0.88, 1.90, 0.70, 1.25), namely 1.0998, an annual average growth of 9.98%. The arithmetic mean of these annual returns is 16.6% per annum, which is not a meaningful average because growth rates do not combine additively.

The geometric mean can be understood in terms ofgeometry. The geometric mean of two numbers,a{\displaystyle a} andb{\displaystyle b}, is the length of one side of asquare whose area is equal to the area of arectangle with sides of lengthsa{\displaystyle a} andb{\displaystyle b}. Similarly, the geometric mean of three numbers,a{\displaystyle a},b{\displaystyle b}, andc{\displaystyle c}, is the length of one edge of acube whose volume is the same as that of acuboid with sides whose lengths are equal to the three given numbers.

The geometric mean is one of the three classicalPythagorean means, together with the arithmetic mean and theharmonic mean. For all positive data sets containing at least one pair of unequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between (seeInequality of arithmetic and geometric means.)

Formulation

[edit]

The geometric mean of a data set{a1,a2,…,an}{\textstyle \left\{a_{1},a_{2},\,\ldots ,\,a_{n}\right\}} is given by:

(∏i=1nai)1n=a1a2⋯antn.{\displaystyle {\biggl (}\prod _{i=1}^{n}a_{i}{\biggr )}^{\frac {1}{n}}={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}.}[3]

That is, thenth root of theproduct of the elements. For example, for1,2,3,4{\textstyle 1,2,3,4}, the product1⋅2⋅3⋅4{\textstyle 1\cdot 2\cdot 3\cdot 4} is24{\textstyle 24}, and the geometric mean is the fourth root of 24, approximately 2.213.

Formulation using logarithms

[edit]

The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms.[4] By usinglogarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:

Whena1,a2,…,an>0{\displaystyle a_{1},a_{2},\dots ,a_{n}>0}

(∏i=1nai)1n=exp⁡(1n∑i=1nln⁡ai),{\displaystyle {\biggl (}\prod _{i=1}^{n}a_{i}{\biggr )}^{\frac {1}{n}}=\exp {\biggl (}{\frac {1}{n}}\sum _{i=1}^{n}\ln a_{i}{\biggr )},}

since|ln⁡a1a2⋯antn=1nln⁡(a1a2⋯an)=1n(ln⁡a1+ln⁡a2+⋯+ln⁡an).{\displaystyle \textstyle {\vphantom {\Big |}}\ln {\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}{\vphantom {t}}}}={\frac {1}{n}}\ln(a_{1}a_{2}\cdots a_{n})={\frac {1}{n}}(\ln a_{1}+\ln a_{2}+\cdots +\ln a_{n}).}

This is sometimes called thelog-average (not to be confused with thelogarithmic average). It is simply thearithmetic mean of the logarithm-transformed values ofai{\displaystyle a_{i}} (i.e., the arithmetic mean on the log scale), using the exponentiation to return to the original scale, i.e., it is thegeneralized f-mean withf(x)=log⁡x{\displaystyle f(x)=\log x}. A logarithm of any base can be used in place of the natural logarithm. For example, the geometric mean of⁠1{\displaystyle 1}⁠,⁠2{\displaystyle 2}⁠,⁠8{\displaystyle 8}⁠, and⁠16{\displaystyle 16}⁠ can be calculated using logarithms base 2:

1⋅2⋅8⋅164=2(log21+log22+log28+log216)/4=2(0+1+3+4)/4=22=4.{\displaystyle {\sqrt[{4}]{1\cdot 2\cdot 8\cdot 16}}=2^{(\log _{2}\!1\,+\,\log _{2}\!2\,+\,\log _{2}\!8\,+\,\log _{2}\!16)/4}=2^{(0\,+\,1\,+\,3\,+\,4)/4}=2^{2}=4.}

Related to the above, it can be seen that for a given sample of pointsa1,…,an{\displaystyle a_{1},\ldots ,a_{n}}, the geometric mean is the minimizer of

f(a)=∑i=1n(log⁡ai−log⁡a)2=∑i=1n(log⁡aia)2{\displaystyle f(a)=\sum _{i=1}^{n}(\log a_{i}-\log a)^{2}=\sum _{i=1}^{n}\left(\log {\frac {a_{i}}{a}}\right)^{2}},

whereas the arithmetic mean is the minimizer of

f(a)=∑i=1n(ai−a)2{\displaystyle f(a)=\sum _{i=1}^{n}(a_{i}-a)^{2}}.

Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).

In computer implementations, naïvely multiplying many numbers together can causearithmetic overflow orunderflow. Calculating the geometric mean using logarithms is one way to avoid this problem.

Related concepts

[edit]

Iterative means

[edit]

The geometric mean of a data setis less than the data set'sarithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of thearithmetic-geometric mean, an intersection of the two which always lies in between.

The geometric mean is also thearithmetic-harmonic mean in the sense that if twosequences (an{\textstyle a_{n}}) and (hn{\textstyle h_{n}}) are defined:

an+1=an+hn2,a0=x{\displaystyle a_{n+1}={\frac {a_{n}+h_{n}}{2}},\quad a_{0}=x}

and

hn+1=2anhnan+hn,h0=y{\displaystyle h_{n+1}={\frac {2{a_{n}}{h_{n}}}{a_{n}+h_{n}}},\quad h_{0}=y}

wherehn+1{\textstyle h_{n+1}} is theharmonic mean of the previous values of the two sequences, thenan{\textstyle a_{n}} andhn{\textstyle h_{n}} will converge to the geometric mean ofx{\textstyle x} andy{\textstyle y}. The sequences converge to a common limit, and the geometric mean is preserved:

ai+1hi+1=ai+hi22aihiai+hi=aihi{\displaystyle {\sqrt {a_{i+1}h_{i+1}}}={\sqrt {{\frac {a_{i}+h_{i}}{2}}{\frac {2{a_{i}}{h_{i}}}{a_{i}+h_{i}}}}}={\sqrt {{a_{i}}{h_{i}}}}}

Replacing the arithmetic and harmonic mean by a pair ofgeneralized means of opposite, finite exponents yields the same result.

Comparison to arithmetic mean

[edit]
Proof without words of theAM–GM inequality:
PR is the diameter of a circle centered on O; its radius AO is thearithmetic mean ofa andb. Triangle PGR is aright triangle fromThales's theorem, enabling use of thegeometric mean theorem to show that itsaltitude GQ is thegeometric mean. For any ratioa:b,AO ≥ GQ.
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]
Main article:Inequality of arithmetic and geometric means

The geometric mean of a non-empty data set of positive numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to amean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.[5]

Geometric mean of a continuous function

[edit]

Iff:[a,b]→(0,∞){\displaystyle f:[a,b]\to (0,\infty )} is a positive continuous real-valued function, its geometric mean over this interval is

GM[f]=exp⁡(1b−a∫abln⁡f(x)dx){\displaystyle {\text{GM}}[f]=\exp \left({\frac {1}{b-a}}\int _{a}^{b}\ln f(x)dx\right)}

For instance, taking the identity functionf(x)=x{\displaystyle f(x)=x} over the unit interval shows that the geometric mean of the positive numbers between 0 and 1 is equal to1e{\displaystyle {\frac {1}{e}}}.

Applications

[edit]

Average proportional growth rate

[edit]
Further information:Compound annual growth rate

The geometric mean is more appropriate than thearithmetic mean for describing proportional growth, bothexponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as thecompound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

As an example, suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, for growth rates of 80%, 16.7% and 42.9% respectively. Using thearithmetic mean calculates a (linear) average growth of 46.5% (calculated by(80%+16.7%+42.9%)÷3{\displaystyle (80\%+16.7\%+42.9\%)\div 3}). However, when applied to the 100 orange starting yield, 46.5% annual growth results in 314 oranges after three years of growth, rather than the observed 300. The linear average overstates the rate of growth.

Instead, using the geometric mean, the average yearly growth is approximately 44.2% (calculated by1.80×1.167×1.4293{\displaystyle {\sqrt[{3}]{1.80\times 1.167\times 1.429}}}). Starting from a 100 orange yield with annual growth of 44.2% gives the expected 300 orange yield after three years.

In order to determine the average growth rate, it is not necessary to take the product of the measured growth rates at every step. Let the quantity be given as the sequencea0,a1,...,an{\displaystyle a_{0},a_{1},...,a_{n}}, wheren{\displaystyle n} is the number of steps from the initial to final state. The growth rate between successive measurementsak{\displaystyle a_{k}} andak+1{\displaystyle a_{k+1}} isak+1/ak{\displaystyle a_{k+1}/a_{k}}. The geometric mean of these growth rates is then just:

(a1a0a2a1⋯anan−1)1n=(ana0)1n.{\displaystyle \left({\frac {a_{1}}{a_{0}}}{\frac {a_{2}}{a_{1}}}\cdots {\frac {a_{n}}{a_{n-1}}}\right)^{\frac {1}{n}}=\left({\frac {a_{n}}{a_{0}}}\right)^{\frac {1}{n}}.}

Normalized values

[edit]

The fundamental property of the geometric mean, which does not hold for any other mean, is that for two sequencesX{\displaystyle X} andY{\displaystyle Y} of equal length,

GM⁡(XiYi)=GM⁡(Xi)GM⁡(Yi){\displaystyle \operatorname {GM} \left({\frac {X_{i}}{Y_{i}}}\right)={\frac {\operatorname {GM} (X_{i})}{\operatorname {GM} (Y_{i})}}}.

This makes the geometric mean the only correct mean when averagingnormalized results; that is, results that are presented as ratios to reference values.[6] This is the case when presenting computer performance with respect to a reference computer, or when computing a single average index from several heterogeneous sources (for example, life expectancy, education years, and infant mortality). In this scenario, using the arithmetic or harmonic mean would change the ranking of the results depending on what is used as a reference. For example, take the following comparison of execution time of computer programs:

Table 1

 Computer AComputer BComputer C
Program 111020
Program 2100010020
Arithmetic mean500.55520
Geometric mean31.622...31.622...20
Harmonic mean1.998...18.182...20

The arithmetic and geometric means "agree" that computer C is the fastest. However, by presenting appropriately normalized valuesand using the arithmetic mean, we can show either of the other two computers to be the fastest. Normalizing by A's result gives A as the fastest computer according to the arithmetic mean:

Table 2

 Computer AComputer BComputer C
Program 111020
Program 210.10.02
Arithmetic mean15.0510.01
Geometric mean110.632...
Harmonic mean10.198...0.039...

while normalizing by B's result gives B as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

Table 3

 Computer AComputer BComputer C
Program 10.112
Program 21010.2
Arithmetic mean5.0511.1
Geometric mean110.632
Harmonic mean0.198...10.363...

and normalizing by C's result gives C as the fastest computer according to the arithmetic mean but A as the fastest according to the harmonic mean:

Table 4

 Computer AComputer BComputer C
Program 10.050.51
Program 25051
Arithmetic mean25.02525.251
Geometric mean1.581...51
Harmonic mean0.099...0.909...1

In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.

However, this reasoning has been questioned.[7]Giving consistent results is not always equal to giving the correct results. In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time (using the arithmetic mean), and then normalize that result to one of the computers. The three tables above just give a different weight to each of the programs, explaining the inconsistent results of the arithmetic and harmonic means (Table 4 gives equal weight to both programs, the Table 2 gives a weight of 1/1000 to the second program, and the Table 3 gives a weight of 1/100 to the second program and 1/10 to the first one). The use of the geometric mean for aggregating performance numbers should be avoided if possible, because multiplying execution times has no physical meaning, in contrast to adding times as in the arithmetic mean. Metrics that are inversely proportional to time (speedup,IPC) should be averaged using the harmonic mean.

The geometric mean can be derived from thegeneralized mean as its limit asp{\displaystyle p} goes to zero. Similarly, this is possible for the weighted geometric mean.

Financial

[edit]

The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past theFT 30 index used a geometric mean.[8] It is also used in theCPI calculation[9] and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.

This has the effect of understating movements in the index compared to using the arithmetic mean.[8]

Applications in the social sciences

[edit]

Although the geometric mean has been relatively rare in computing social statistics, starting from 2010 the United Nations Human Development Index did switch to this mode of calculation, on the grounds that it better reflected the non-substitutable nature of the statistics being compiled and compared:

The geometric mean decreases the level of substitutability between dimensions [being compared] and at the same time ensures that a 1 percent decline in say life expectancy at birth has the same impact on the HDI as a 1 percent decline in education or income. Thus, as a basis for comparisons of achievements, this method is also more respectful of the intrinsic differences across the dimensions than a simple average.[10]

Not all values used to compute theHDI (Human Development Index) are normalized; some of them instead have the form(X−Xmin)/(Xnorm−Xmin){\displaystyle \left(X-X_{\text{min}}\right)/\left(X_{\text{norm}}-X_{\text{min}}\right)}. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

The equally distributed welfare equivalent income associated with anAtkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is anLp norm divided by the number of elements, with p equal to one minus the inequality aversion parameter.

Geometry

[edit]
The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. UsingPythagoras' theorem on the 3 triangles of sides(p + q,r,s ),(r,p,h ) and(s,h,q ),
(p+q)2=r2+s2p2+2pq+q2=p2+h2⏞+h2+q2⏞2pq=2h2∴h=pq{\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}

In the case of aright triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as thegeometric mean theorem.

In anellipse, thesemi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from afocus; it is also the geometric mean of thesemi-major axis and thesemi-latus rectum. Thesemi-major axis of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to eitherdirectrix.

Another way to think about it is as follows:

Consider a circle with radiusr{\displaystyle r}. Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengthsa{\displaystyle a} andb{\displaystyle b}.

Since the area of the circle and the ellipse stays the same, we have:

πr2=πabr2=abr=ab{\displaystyle {\begin{aligned}\pi r^{2}&=\pi ab\\r^{2}&=ab\\r&={\sqrt {ab}}\end{aligned}}}

The radius of the circle is the geometric mean of the semi-major and the semi-minor axes of the ellipse formed by deforming the circle.

Distance to thehorizon of asphere (ignoring theeffect of atmospheric refraction when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

The geometric mean is used in both in the approximation ofsquaring the circle by S.A. Ramanujan[11] and in the construction of theheptadecagon with "mean proportionals".[12]

Aspect ratios

[edit]
Equal area comparison of the aspect ratios used by Kerns Powers to derive theSMPTE16:9 standard.[13]  TV 4:3/1.33 in red,  1.66 in orange,  16:9/1.77 in blue,  1.85 in yellow,  Panavision/2.2 in mauve and  CinemaScope/2.35 in purple.

The geometric mean has been used in choosing a compromiseaspect ratio in film and video: given two aspect ratios, the geometric mean of them provides a compromise between them, distorting or cropping both in some sense equally. Concretely, two equal area rectangles (with the same center and parallel sides) of different aspect ratios intersect in a rectangle whose aspect ratio is the geometric mean, and their hull (smallest rectangle which contains both of them) likewise has the aspect ratio of their geometric mean.

Inthe choice of 16:9 aspect ratio by theSMPTE, balancing 2.35 and 4:3, the geometric mean is2.35×43≈1.7701{\textstyle {\sqrt {2.35\times {\frac {4}{3}}}}\approx 1.7701}, and thus16:9=1.777¯{\textstyle 16:9=1.77{\overline {7}}}... was chosen. This was discoveredempirically by Kerns Powers, who cut out rectangles with equal areas and shaped them to match each of the popular aspect ratios. When overlapped with their center points aligned, he found that all of those aspect ratio rectangles fit within an outer rectangle with an aspect ratio of 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1.[13] The value found by Powers is exactly the geometric mean of the extreme aspect ratios,4:3 (1.33:1) andCinemaScope (2.35:1), which is coincidentally close to16:9{\textstyle 16:9} (1.777¯:1{\textstyle 1.77{\overline {7}}:1}). The intermediate ratios have no effect on the result, only the two extreme ratios.

Applying the same geometric mean technique to 16:9 and 4:3 approximately yields the14:9 (1.555¯{\textstyle 1.55{\overline {5}}}...) aspect ratio, which is likewise used as a compromise between these ratios.[14] In this case 14:9 is exactly thearithmetic mean of16:9{\textstyle 16:9} and4:3=12:9{\textstyle 4:3=12:9}, since 14 is the average of 16 and 12, while the precisegeometric mean is169×43≈1.5396≈13.8:9,{\textstyle {\sqrt {{\frac {16}{9}}\times {\frac {4}{3}}}}\approx 1.5396\approx 13.8:9,} but the two differentmeans, arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).

Paper formats

[edit]

The geometric mean is also used to calculate B and C seriespaper formats. TheBn{\displaystyle B_{n}} format has an area which is the geometric mean of the areas ofAn{\displaystyle A_{n}} andAn−1{\displaystyle A_{n-1}}. For example, the area of a B1 paper is22m2{\textstyle {\frac {\sqrt {2}}{2}}\mathrm {m} ^{2}}, because it is the geometric mean of the areas of an A0 (1m2{\textstyle 1\mathrm {m} ^{2}}) and an A1 (12m2{\textstyle {\frac {1}{2}}\mathrm {m} ^{2}}) paper(1m2⋅12m2=12m4={\textstyle {\sqrt {1\mathrm {m} ^{2}\cdot {\frac {1}{2}}\mathrm {m} ^{2}}}={\sqrt {{\frac {1}{2}}\mathrm {m} ^{4}}}={}}​12m2=22m2{\textstyle {\frac {1}{\sqrt {2}}}\mathrm {m} ^{2}={\frac {\sqrt {2}}{2}}\mathrm {m} ^{2}}).

The same principle applies with the C series, whose area is the geometric mean of the A and B series. For example, the C4 format has an area which is the geometric mean of the areas of A4 and B4.

An advantage that comes from this relationship is that an A4 paper fits inside a C4 envelope, and both fit inside a B4 envelope.

Other applications

[edit]

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. ^Matt Friehauf, Mikaela Hertel, Juan Liu, and Stacey Luong"On Compass and Straightedge Constructions: Means"(PDF). UNIVERSITY of WASHINGTON, DEPARTMENTOF MATHEMATICS. 2013. Retrieved14 June 2018.
  2. ^David E. Joyce, ed. (2013)."Euclid, Book VI, Proposition 13". Clark University. Retrieved19 July 2019.
  3. ^"2.5: Geometric Mean".Statistics LibreTexts. 2019-04-20. Retrieved2021-08-16.
  4. ^Crawley, Michael J. (2005).Statistics: An Introduction using R. John Wiley & Sons Ltd.ISBN 9780470022986.
  5. ^Mitchell, Douglas W. (2004). "More on spreads and non-arithmetic means".The Mathematical Gazette.88 (511):142–144.doi:10.1017/S0025557200174534.S2CID 168239991.
  6. ^Fleming, Philip J.; Wallace, John J. (1986)."How not to lie with statistics: the correct way to summarize benchmark results".Communications of the ACM.29 (3):218–221.doi:10.1145/5666.5673.S2CID 1047380.
  7. ^Smith, James E. (1988)."Characterizing computer performance with a single number".Communications of the ACM.31 (10):1202–1206.doi:10.1145/63039.63043.S2CID 10805363.
  8. ^abRowley, Eric E. (1987).The Financial System Today. Manchester University Press.ISBN 0719014875.
  9. ^"Measuring price inflation"(PDF). Government Actury's Department. March 2017. Retrieved15 July 2023 – via gov.uk.
  10. ^"Frequently Asked Questions - Human Development Reports".hdr.undp.org.Archived from the original on 2011-03-02.
  11. ^Ramanujan, S. (1914)."Modular equations and approximations toπ"(PDF).Quarterly Journal of Mathematics.45:350–372.
  12. ^T.P. StowellExtract from Leybourn's Math. Repository, 1818 inThe Analyst viaGoogle Books
  13. ^ab"TECHNICAL BULLETIN: Understanding Aspect Ratios"(PDF). The CinemaSource Press. 2001.Archived(PDF) from the original on 2009-09-09. Retrieved2009-10-24.
  14. ^US 5956091, "Method of showing 16:9 pictures on 4:3 displays", issued September 21, 1999 
  15. ^MacEvoy, Bruce."Colormaking Attributes: Measuring Light & Color".handprint.com/LS/CVS/color.html. Colorimetry.Archived from the original on 2019-07-14. Retrieved2020-01-02.
  16. ^Henry Ludwell Moore (1895).Von Thünen's Theory of Natural Wages. G. H. Ellis.

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 (see alsoTemplate:Least squares and 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
Authority control databasesEdit this at Wikidata
Retrieved from "https://en.wikipedia.org/w/index.php?title=Geometric_mean&oldid=1316221500"
Category:
Hidden categories:
[8]ページ先頭
©2009-2025 Movatter.jp