# Geometric mean

geometric averagegeometricmeanGeometric returngeometrical meangeometrical meansmathematic geometric meanmean proportional
In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum).wikipedia
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### Arithmetic mean

meanaveragearithmetic
In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). It is simply computing the arithmetic mean of the logarithm-transformed values of a_i (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with.
The term "arithmetic mean" is preferred in some contexts in mathematics and statistics because it helps distinguish it from other means, such as the geometric mean and the harmonic mean.

### Average

Rushing averageReceiving averagemean
In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum).
The geometric mean of n positive numbers is obtained by multiplying them all together and then taking the nth root.

### Mean

mean valuepopulation meanaverage
In mathematics, the geometric mean is a mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum).
The geometric mean is an average that is useful for sets of positive numbers that are interpreted according to their product and not their sum (as is the case with the arithmetic mean); e.g., rates of growth.

### Harmonic mean

harmonicweighted harmonic meanharmonic average
The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.
it is more apparent that the harmonic mean is related to the arithmetic and geometric means.

### Inequality of arithmetic and geometric means

arithmetic-geometric mean inequalityAM-GM inequalityAM–GM inequality
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 (see Inequality of arithmetic and geometric means.)
In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

### Generalized mean

power meangeneralisedHölder generalized mean
Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.
In mathematics, generalized means are a family of functions for aggregating sets of numbers, that include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means).

### Pythagorean means

The geometric mean is also one of the three classical Pythagorean means, together with the aforementioned arithmetic mean and the harmonic mean.
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians because of their importance in geometry and music.

### Quasi-arithmetic mean

generalised f-meangeneralized ''f''-meangeneralized ƒ-mean
It is simply computing the arithmetic mean of the logarithm-transformed values of a_i (i.e., the arithmetic mean on the log scale) and then using the exponentiation to return the computation to the original scale, i.e., it is the generalised f-mean with.
In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov.

### Human Development Index

HDIhuman developmentstandard of living
Not all values used to compute the HDI (Human Development Index) are normalized; some of them instead have the form.
Finally, the HDI is the geometric mean of the previous three normalized indices:

### Logarithmic mean

logarithmic average
This is sometimes called the log-average (not to be confused with the logarithmic average).
The logarithmic mean of two numbers is smaller than the arithmetic mean but larger than the geometric mean (unless the numbers are the same, in which case all three means are equal to the numbers):

### Square root

For instance, the geometric mean of two numbers, say 2 and 8, is just the square root of their product, that is,.
In his Elements, Euclid (fl. 300 BC) gave the construction of the geometric mean of two quantities in two different places: Proposition II.14 and Proposition VI.13.

### Arithmetic–geometric mean

arithmetic-geometric meanAGMarithmetic geometric mean
This allows the definition of the arithmetic-geometric mean, an intersection of the two which always lies in between.
The arithmetic–harmonic mean can be similarly defined, but takes the same value as the geometric mean.

### Logarithm

logarithmsloglogarithmic function
For example, the geometric mean of 2 and 8 can be calculated as the following, where b is any base of a logarithm (commonly 2, e or 10):
(arithmetic mean) and \sqrt{xy} (geometric mean) of and then let those two numbers become the next and.

### FT 30

Financial News'' 30-share indexFT 30 indexFT Index
For example, in the past the FT 30 index used a geometric mean.
The FT 30 index was calculated using the geometric mean.

### Right triangle

right-angled triangleright angled triangleright angle triangle
In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex.
The altitude to the hypotenuse is the geometric mean (mean proportional) of the two segments of the hypotenuse.

### Hyperbolic coordinates

hyperbolic rotation
Hyperbolic coordinates
The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y.

### Semi-major and semi-minor axes

semi-major axissemimajor axissemi-major axes
In an ellipse, the semi-minor axis is the geometric mean of the maximum and minimum distances of the ellipse from a focus; it is also the geometric mean of the semi-major axis and the semi-latus rectum.
The semi-minor axis of an ellipse is the geometric mean of these distances:

### Geometric mean filter

The geometric mean filter is used as a noise filter in image processing.
It is based on the mathematic geometric mean.

### Multiplicative calculus

Non-Newtonian calculusnon-Newtonian calculi Non-Newtonian calculus
Multiplicative calculus
Furthermore, just as the arithmetic average (of functions) is the "natural" average in the classical calculus, the well-known geometric average is the "natural" average in the geometric calculus.

### Geometric mean theorem

Euclid's geometric mean theoremmean proportionalright triangle altitude theorem
Geometric mean theorem
It states that the geometric mean of the two segments equals the altitude.

### Geometric standard deviation

Geometric standard deviation
In probability theory and statistics, the geometric standard deviation describes how spread out are a set of numbers whose preferred average is the geometric mean.

### Spectral flatness

In signal processing, spectral flatness, a measure of how flat or spiky a spectrum is, is defined as the ratio of the geometric mean of the power spectrum to its arithmetic mean.
The spectral flatness is calculated by dividing the geometric mean of the power spectrum by the arithmetic mean of the power spectrum, i.e.:

### Anti-reflective coating

antireflection coatinganti-reflection coatinganti-reflective
In optical coatings, where reflection needs to be minimised between two media of refractive indices n 0 and n 2, the optimum refractive index n 1 of the anti-reflective coating is given by the geometric mean:.
This optimal value is given by the geometric mean of the two surrounding indices:

### Log-normal distribution

lognormallog-normallognormal distribution
Log-normal distribution
Let denote the geometric mean, and the geometric standard deviation of the random variable X, and let and be the arithmetic mean, or expected value, and the arithmetic standard deviation as usual.

### Squaring the circle

square the circlequadrature of the circlesquaring of the circle
Both in the approximation of squaring the circle according to S.A. Ramanujan (1914) and in the construction of the Heptadecagon according to "sent by T. P. Stowell, credited to Leybourn's Math. Repository, 1818", the geometric mean is employed.
Extend beyond A and beat the circular arc b 1 around O with radius, it results S'. Bisect the line segment in D and draw the semicircle b 2 over D. Draw a straight line from O through C up to the semicircle b 2, it cuts b 2 in E. The line segment is the mean proportional between and, also called geometric mean.