Geometric mean

The geometric mean of the lengths l ₁ and l ₂ is the length l _g . In this example l _{2 is} perpendicular to l ₁ at point B ; See animation here .

The geometric mean or the mean proportional is the mean value that is obtained from the product of the positive numbers considered using the -th root . The geometric mean is always less than or equal to the arithmetic mean . It is used, among other things, in statistics , finance and also in geometric constructions, such as those used in B. are listed in application examples . ${\displaystyle n}$ ${\displaystyle n}$

The two numbers 1 and 2, for example, have the geometric mean     (arithmetic mean = 1.5; the larger number, here: 2, is rated lower in the geometric mean). ${\displaystyle {\sqrt[{2}]{1\cdot 2}}\approx 1{,}41\ ,}$

definition

The geometric mean of the numbers (with for all ) is given by the -th root of the product of the numbers: ${\displaystyle n}$ ${\displaystyle x_{1},x_{2},\dotsc ,x_{n}}$ ${\displaystyle x_{i}>0}$ ${\displaystyle i=1,\ldots ,n}$ ${\displaystyle n}$ ${\displaystyle n}$

${\displaystyle {\bar {x}}_{\mathrm {geom} }={\sqrt[{n}]{\prod _{i=1}^{n}{x_{i}}}}={\sqrt[{n}]{x_{1}\cdot x_{2}\dotsm x_{n}}}}$

Analogous to the weighted arithmetic mean , a weighted geometric mean is defined with weights : ${\displaystyle w_{i}>0}$

${\displaystyle {\bar {x}}_{\mathrm {geom} }=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{\frac {1}{w}}={\sqrt[{w}]{\prod _{i=1}^{n}x_{i}^{w_{i}}}}}$, ${\displaystyle \textstyle w:=\sum _{i=1}^{n}w_{i}}$

properties

In contrast to the arithmetic mean , the geometric mean is only defined for non-negative numbers and mostly only makes sense for really positive real numbers , because if one factor is equal to zero , the entire product is already zero. It is not used for complex numbers because the complex roots are ambiguous. ${\displaystyle x_{i}}$

The inequality of the arithmetic and geometric mean says that

${\displaystyle {\bar {x}}_{\mathrm {geom} }\leq {\bar {x}}_{\mathrm {arithm} }}$,

so that the geometric mean is never greater than the arithmetic mean.

The logarithm of the geometric mean is the arithmetic mean of the logarithms, whereby the base of the logarithm can be chosen arbitrarily: ${\displaystyle a}$

${\displaystyle \log _{a}{\bar {x}}_{\mathrm {geom} }={\frac {1}{n}}\sum _{i=1}^{n}\log _{a}x_{i},}$

which results in a practicable calculation method for large . ${\displaystyle n}$

The arithmetic-geometric mean is a number that lies between the arithmetic and geometric mean.

Also applies to and ${\displaystyle n=2}$ ${\displaystyle w_{1}=w_{2}=1}$

${\displaystyle x_{\mathrm {geom} }={\sqrt {x_{\mathrm {arithm} }\cdot x_{\mathrm {harm} }}}}$

with the arithmetic and harmonic mean .

Geometric interpretations

• As shown above, the Thales circle creates a right-angled triangle AC'E. With the help of the height theorem , we can then calculate what exactly corresponds to the formula for the geometric mean. ${\displaystyle l_{g}}$ ${\displaystyle l_{g}={\sqrt {l_{a}\cdot l_{b}}}}$

• The geometric mean of two numbers and gives the side length of a square that has the same area as the rectangle with side lengths and . This fact is illustrated by the geometric squaring of the rectangle . ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle b}$

• In the same way, the geometric mean for three numbers corresponds to the side length of a cube, which has the same volume as the cuboid with the three side lengths, and in -dimensionally for numbers the side lengths of hypercubes . ${\displaystyle n}$ ${\displaystyle n}$

Application examples

When geometrically averaging two values, both values ​​deviate from the mean by the same factor. This is not the case with the arithmetic mean. The arithmetic mean 5 results from 1 and 9. The 1 is removed from the mean value 5 by a factor of 5, while the 9 is only a factor of 1.8 away from it. The geometric mean from 1 and 9, on the other hand, gives the mean value 3. Both the low value 1 and the high value 9 are away from the mean value 3 by a factor of 3. The difference between the arithmetic and geometric mean can be considerable, which in practice can lead to the misinterpretation of averages. For example, 0.02 and 10 result in the mean values ​​5.01 (arithmetic) and 0.45 (geometric).

Examples:

• The geometric mean of two values is e.g. As of and : . ${\displaystyle a,b}$ ${\displaystyle {\sqrt {ab}}}$ ${\displaystyle a=3}$ ${\displaystyle b=300}$ ${\displaystyle {\sqrt {3\cdot 300}}=30}$
• Properties are determined from a 0.1 molar solution and a 10 molar solution, which change depending on the concentration following a linear relationship. In order to obtain a solution that has average properties, the geometric mean must be formed, which in this case = 1. The arithmetic mean, on the other hand, would describe a 5.05 molar solution, which primarily has the properties of the 10 molar solution, i.e. it does not behave average at all.
• The golden ratio is based on the geometric mean.
• The geometric mean is used both in the approximate construction of the quadrature of the circle according to SA Ramanujan (1914) and in the construction of the seventeenth-corner from 1818 (seventeen-corner / see also) .
• A credit will bear interest in the first year with two percent in the second year with seven and the third year of five percent. Which interest rate constant over the three years would have resulted in the same capital in the end? ${\displaystyle G}$ ${\displaystyle p}$

Balance at the end of the third year: ${\displaystyle G_{\mathrm {Ende} }}$

${\displaystyle G_{\mathrm {Ende} }=\left(1+{\frac {2}{100}}\right)\left(1+{\frac {7}{100}}\right)\left(1+{\frac {5}{100}}\right)G}$

or written with interest factors

${\displaystyle G_{\mathrm {Ende} }=1{,}02\cdot 1{,}07\cdot 1{,}05\cdot G}$

With a constant interest rate and the associated interest factor, the result is a credit of ${\displaystyle p}$ ${\displaystyle 1+p}$

${\displaystyle G_{\mathrm {konst} }=(1+p)^{3}\;G}$

With results ${\displaystyle G_{\mathrm {konst} }=G_{\mathrm {Ende} }}$

${\displaystyle (1+p)^{3}G=1{,}02\cdot 1{,}07\cdot 1{,}05\cdot G}$

and the average interest factor calculated to ${\displaystyle 1+p}$

${\displaystyle 1+p={\sqrt[{3}]{1{,}02\cdot 1{,}07\cdot 1{,}05}}\approx 1{,}04646}$

The average interest rate is therefore approx . In general, the average interest factor is calculated from the geometric mean of the interest factors for the individual years. Because of the inequality of the arithmetic and geometric mean , the average interest rate is less than or at best equal to the arithmetic mean of the interest rates, which is in this example . The mean interest factor is calculated as the geometric mean; the mean interest rate can be represented as the f-mean (see f-mean ). ${\displaystyle 4{,}646\,\%}$ ${\displaystyle {\tfrac {14}{3}}\,\%\approx 4{,}667\,\%}$

statistics

In statistics , mean values of absolute frequencies or relative frequencies can be calculated using the weighted geometric mean.

When using relative frequencies , these are used as weights. The following then applies: from which it follows ${\displaystyle \sum _{i=1}^{n}w_{i}=1}$

${\displaystyle {\bar {x}}_{\mathrm {geom} }=\prod _{i=1}^{n}x_{i}^{w_{i}}}$.

If absolute frequencies are used as weights, the mean value is obtained

${\displaystyle {\bar {x}}_{\mathrm {geom} }={\sqrt[{w}]{\prod _{i=1}^{n}x_{i}^{w_{i}}}},\;w=\sum _{i=1}^{n}w_{i}}$.

Holder means

Without weighting

The geometric mean results as a special case of the Hölder mean for . ${\displaystyle p\to 0}$

The definition of the (unweighted) Holder-agent for are: . ${\displaystyle p\to 0}$ ${\displaystyle \lim _{p\to 0}\left({\frac {1}{n}}\sum _{i=1}^{n}{x_{i}^{p}}\right)^{\frac {1}{p}}}$

We can now transform and with the help of de l'Hospital's rule we finally get

${\displaystyle \exp {\left(\lim _{p\to 0}{\frac {\ln {\left[{\frac {1}{n}}\sum _{i=1}^{n}{x_{i}^{p}}\right]}}{p}}\right)}}$.

The logarithmic laws simplify the exponent to .. ${\displaystyle \ln {\left[{\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}\right]}}$

We start in the original term and get the definition of the geometric mean

${\displaystyle \exp \left(\ln \left[{\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}\right]\right)\rightarrow {\bar {x}}_{\mathrm {geom} }={\sqrt[{n}]{\prod _{i=1}^{n}x_{i}}}}$.

With weighting

The weighted geometric mean can also be obtained by calculating the limit value of the weighted Hölder mean

${\displaystyle {\bar {x}}={\sqrt[{w}]{\prod _{i=1}^{n}x_{i}^{w_{i}}}}}$.

For this you have to note that you can normalize any weights and (in order to be able to apply the de l´Hôspital rule) have to use them instead . ${\displaystyle {\frac {w_{i}}{\sum _{i=1}^{n}w_{i}}}}$ ${\displaystyle w_{i}}$

The unweighted geometric mean results in turn. ${\displaystyle w_{i}={\frac {1}{n}}}$

See also

• Arithmetic mean
• Harmonious mean
• Holder mean , generalization of the geometric mean
• Inequality of the arithmetic and geometric mean

Web links

• Eric W. Weisstein : Geometric Mean . In: MathWorld (English).
• Calculate the geometric mean of two numbers compared to the arithmetic mean

Individual evidence

1. ↑ Eckard Specht: A.14 The arithmetic mean. University of Magdeburg, accessed April 25, 2020 . 
2. ↑ Euclid : Stoicheia (Euclid's Elements) VI.13. For two lines find the line that is related to them like the middle link in continuously the same proportion. accessed on November 20, 2018
3. ^ Alan Anderson: Business Statistics for Dummies . John Wiley & Sons, 2014, ISBN 978-1-118-78449-5 , pp. 46 .
4. ↑ ^{a b} Geometric mean. In: Mathebibel.de. Retrieved August 17, 2019 . 
5. ↑ Feng Qi: Generalized abstract mean values. P. 1 , accessed on August 17, 2019 (English). 
6. ↑ Feng Qi: Generalized abstract mean values. P. 3 , accessed on August 17, 2019 (English).