Geometric mean

A geometric mean (きかへいきん British: geometric mean) or a geometrical mean is a representative figure of the numerical value group with a kind of the average in the mathematics. Many people hear it with average and resemble an arithmetical mean to remember, but write it not adding each numerical value and is provided by taking 冪根 (if numerical value is n unit n root) of the product.

Table of contents

Summary

If the geometric mean of two numbers is the square root of product and, for example, is 2 and 8 ${\displaystyle {\sqrt[{2}]{2\times 8}} = 4}$ となる. In addition, three number 4 and 1 and geometric mean of 1/32 are cubic roots of those product (1/8),${\displaystyle {\sqrt[{3}]{4\times 1\times 1/32}}=1/2}$ となる.

I can comment on the geometric mean geometrically. The geometric mean of two number a and b is equivalent with the length of the side demanding one side of length of the square of the area same as a rectangle of a and b. Similarly, it is nothing but that it demands length of one side of the regular hexahedron of the volume same as a cuboid to be the head of the side in them to demand a, b, the geometric mean of the number of three called c.

The geometric mean can handle only a positive number [¹]. I often use it for a value and the value with the exponential property that I often multiply each other and, for example, am used for the rate of interest of data and the treasury investment about the growth of the population.

The geometric mean is one of three classic average called "mean (en) of Pythagoras" (as for the others an arithmetical mean and harmonic average). When I found average of the positive numerical value group including the different value, harmonic average is the always smallest, and an arithmetical mean grows biggest, and, as for the geometric mean, it is in the middle.

Calculation

Data meeting ${\displaystyle a_{1},a_{2},\ldots ,a_{n}}$ The の geometric mean is demanded in the next expression.

${\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{1/n}={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}}$

The geometric mean becomes with less than it of arithmetical mean of the same data meeting (only in the case of a value same as for both equalling all numerical value). Therefore the arithmetic geometric mean that mixed both is defined and is always the middle price of an arithmetical mean and the geometric mean.

Two line (a_n) and (h_n)

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

と

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

When it is defined のように, the geometric mean becomes the arithmetic harmonic average, and a_n and h_n converge in a geometric mean of x and y.

This always understands, in fact, (theorem of the Bolzano = Y Ersch trass) geometric mean that two lines converge in a common limit from a fact to be the same easily.

${\displaystyle {\sqrt {a_{i}h_{i}}}={\sqrt {\frac {a_{i}+h_{i}}{\frac {a_{i}+h_{i}}{h_{i}a_{i}}}}}={\sqrt {\frac {a_{i}+h_{i}}{{\frac {1}{a_{i}}}+{\frac {1}{h_{i}}}}}}={\sqrt {a_{i+1}h_{i+1}}}}$

The same result is provided even if I replace an arithmetical mean and harmonic average with one pair of generalization average of the limited index in an inverse number.

with the logarithmic arithmetical mean of relationships

I can express multiplication by the addition when I transform an expression using a logarithmic property and can express べき 乗 by multiplication.

${\displaystyle \left(\prod _{i=1}^{n}a_{i}\right)^{1/n}=\exp \left[{\frac {1}{n}}\sum _{i=1}^{n}\ln a_{i}\right]}$

I call this the logarithm average. Numerical value of the original data group ${\displaystyle a_{i}}$ I convert it into を logarithm and demand an arithmetical mean and I apply an exponential function and get an original numerical geometric mean. It is nothing but this namely f(x) = log x and the generalization average that I did. For example, I can calculate the geometric mean of 2 and 8 as follows.

${\displaystyle b^{\left(\log _{b}(2)+\log _{b}(8)\right)/2}=4}$

b is a logarithmic bottom here, and any value is all right (generally I use 2, e, 10 either).

with an arithmetical mean and the mean conservative diffusion of relationships

When they made mean conservative diffusion [²] on the numerical value group of the different value each, geometric means always become small [³].

Calculation at the regular interval

When use the geometric mean to find a mean growth rate of some kind of quantity; an initial value ${\displaystyle a_{0}}$ と latest value ${\displaystyle a_{n}}$ But, the geometric mean of the latest growth rate is demanded in the next expression if known without using the value on the way.

${\displaystyle \left({\frac {a_{n}}{a_{0}}}\right)^{\frac {1}{n}}}$

Here ${\displaystyle n}$ It is the number of the steps from は 初期値 to the latest state.

Numerical value group ${\displaystyle a_{0},\ldots ,a_{n}}$ とし,${\displaystyle a_{k}}$ と ${\displaystyle a_{k+1}}$ A growth rate between の ${\displaystyle a_{k+1}/a_{k}}$ とする. Then the geometric mean of the growth rate is as follows.

${\displaystyle \left({\frac {a_{1}}{a_{0}}}{\frac {a_{2}}{a_{1}}}\cdots {\frac {a_{n}}{a_{n-1}}}\right)^{\frac {1}{n}}=\left({\frac {a_{n}}{a_{0}}}\right)^{\frac {1}{n}}}$

Use

Growth rate

When I express a growth rate, a geometric mean is suitable than an arithmetical mean even if even exponential growth (English version) (when a growth rate is constant) is not so. I call this yearly average growth rate (CAGR) in the field of business. It is a growth rate when the geometric mean of the growth rate of the period of time grows up in the ratio of uniformity in the period and achieves the same growth.

If I can harvest 100 oranges in the year that there is from the tree of a certain orange and changed with 180, 210, 300 every year afterwards, as for the growth rate of each year after year, it is 80%, 16.7%, 42.9% sequentially. (I divide 80% +16.7% +42.9% by 3) for the arithmetical mean of the growth rate and the mean growth rate become 46.5%. However, 100 oranges are produced in the early years, and it is 314 in the most lifetime if by 46.5% a year grew up afterwards, and it is not to 300. In other words, I greatly estimate a mean growth rate to be when arithmetic averages a growth rate simply.

Alternatively, I can use the geometric mean. 80% of growth rates mean 1.80 times. Therefore when I take 1.80, 1.167, a geometric mean of 1.429 ${\displaystyle {\sqrt[{3}]{1.80\times 1.167\times 1.429}} = 1.443}$ The neighbor, the mean growth rate become 44.3%. If by 44.3% a year grew up for 100 afterwards in the early years, it becomes 300 in the most lifetime.

Applied in the social science

When I calculated social statistics, I rarely used the geometric mean, but came to find the Human Development Index of the United Nations using a geometric mean from 2010. It is said that this reflects properties of the statistic better.

The geometric mean lowers a level of the substitutability between dimensions (compared) and guarantees that 1% of drops of the average life span give Human Development Index influence same as education and the drop of 1% of incomes at the same time at birth. Therefore, it may be said that this expresses the essential difference that crossed the dimension than simple average for the basics of comparison of the achievement degree well [⁴].

Aspect ratio

Various aspect ratio [⁵] of the same area that I showed when Kerns Powers suggested a standard at 16:9 of SMPTE. In 4:3/1.33 of conventional TV, as for red, 1.66, in orange, 16:9/1.77, as for blue, 1.85, in yellow, Panavision /2.2, light purple, CinemaScope/2.35 are purple

The geometric mean has been used for a movie and development of the conciliatory screen aspect ratio of the video. If two aspect ratio takes those geometric means once, I provide the conciliatory aspect ratio that I lent to twist both at the same level, or to cut. Specifically, the domain where they were piled up equals the aspect ratio of the geometric mean of both when I repeat the domains where an area is equal, and vary in aspect ratio so that a side is parallel with the center together. In addition, the domain of the smallest rectangle to all include both becomes the aspect ratio same as a geometric mean.

SMPTE takes a geometric mean of (conventional TV) at 2.35:1 (movie of the scope size) and 4:3 on choosing the aspect ratio called 16:9 and ${\displaystyle {\sqrt {2.35\times {\frac {4}{3}}}}=1.7{\overline {7}}}$ とし, 16:9 = 1.777... I chose を. Kerns Powers arrived at this from experience, and he made the equal rectangle of the area to main aspect ratio and compared it. When I matched those center and repeated it, rectangular aspect ratio to include the whole discovered that it was 1.77:1, and similarly the domain where all rectangles were piled up at the same time discovered that it was in aspect ratio called 1.77:1 [⁵]. The value that Powers discovered is a geometric mean of 4:3 (1.33:1) and 2.35:1 definitely, and 16:9 (1.78:1) is very near. Powers considered the aspect ratio except two, but it is only two of the most extreme shape to affect a geometric mean.

When apply the technique of this geometric mean at 16:9 and 4:3; about 14:9 (1.555...Aspect ratio of) is provided and is used as aspect ratio of the compromise plan equally [⁶]. In this case the geometric mean that, in fact, it is an arithmetical mean of 16:9 and 4:3 at 14:9 (= 12:9 at 4:3 as for the arithmetical mean of 16 and 12 14), and is exact $It is 9} {\displaystyle {\sqrt {{\frac {16}{9}}\times {\frac {4}{3}}}}\approx 1.5396\approx 13.8$ But I can quite consider both the arithmetical mean and the geometric mean equal with a value for a little less than ten minutes.

Spectrum flatness characteristics

The spectrum flatness characteristics in signal processing it express the degree of the flatness of the spectrum, and are defined as a geometric mean of the spectrum density by the ratio of the arithmetical mean.

Geometry

f becomes the geometric mean of d and e

The height when I assumed the oblique side the base of the right-angled triangle is equal to the geometric mean of each segment of a line when I divided the oblique side in the perpendicular line which I drew on the oblique side from the corner that is a right angle.

A semiminor axis is a geometric mean of the maximum and the minimum of the distance with the points in the laps of the ovals in an oval from the focus. On the other hand, the semimajor axis is center point and a geometric mean of the distance with either focus and the distance with center point and the directrix.

Footnote, source

[Help]

1. Because when the ^ product becomes the minus number, 冪根 becomes the imaginary number. In addition, when 0 is included for numerical value, the product always becomes 0, and the geometric mean becomes 0, too.
2. Scatter plural elements of the ^ numerical value group not to change an arithmetical mean
3. ^ Mitchell, Douglas W., "More on spreads and non-arithmetic means," The Mathematical Gazette 88, March 2004, 142-144.
4. ^ FAQ - HUMAN DEVELOPMENT REPORT
5. ^ ^{a b} TECHNICAL BULLETIN: Understanding Aspect Ratios. The CinemaSource Press. (2001). http://www.cinemasource.com/articles/aspect_ratios.pdf#page=8 October 24, 2009 reading. . 
6. It is 1999 in 21 in displays", issued September at pictures on 4:3 at ^ US 5956091, "Method of showing 16:9 

Allied item

• Arithmetical mean
• Arithmetic geometric mean
• Average
• Lognormality distribution
• The square average square root
• Return on investment

Outside link

• Calculation of the geometric mean of two numbers in comparison to the arithmetic solution
• Arithmetic and geometric means
• When to use the geometric mean
• Practical solutions for calculating geometric mean with different kinds of data
• Geometric Mean on MathWorld
• Geometric Meaning of the Geometric Mean
• Geometric Mean Calculator for larger data sets
• Computing Congressional Apportionment using Geometric Mean

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