The

**geometric mean**, in

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, is a type of

meanIn statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

or

averageIn mathematics, an average, or central tendency of a data set is a measure of the "middle" value of the data set. Average is one form of central tendency. Not all central tendencies should be considered definitions of average....

, which indicates the central tendency or typical value of a set of numbers. It is similar to the

arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

, except that the numbers are multiplied and then the

*n*th rootIn mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

(where n is the count of numbers in the set) of the resulting

productIn mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication...

is taken.

For instance, the geometric mean of two numbers, say 2 and 8, is just the

square rootIn mathematics, a square root of a number x is a number r such that r2 = x, or, in other words, a number r whose square is x...

of their product; that is . As another example, the geometric mean of the three numbers 4, 1, and 1/32 is the

cube root of their product (1/8), which is 1/2; that is .

More generally, if the numbers are

, the geometric mean

satisfies

and hence

The latter expression states that the log of the geometric mean is the arithmetic mean of the logs of the numbers.

The geometric mean can also be understood in terms of

geometryGeometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

. The geometric mean of two numbers,

*a* and

*b*, is the length of one side of a

squareIn geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

whose area is equal to the area of a

rectangleIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...

with sides of lengths

*a* and

*b*. Similarly, the geometric mean of three numbers,

*a*,

*b*, and

*c*, is the length of one side of a

cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

whose volume is the same as that of a

cuboidIn geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...

with sides whose lengths are equal to the three given numbers.

The geometric mean applies only to positive numbers. It is also often used for a set of numbers whose values are meant to be multiplied together or are exponential in nature, such as data on the growth of the

human populationThe world population is the total number of living humans on the planet Earth. As of today, it is estimated to be billion by the United States Census Bureau...

or interest rates of a financial investment.

The geometric mean is also one of the three classic

Pythagorean meansIn mathematics, the three classical Pythagorean means are the arithmetic mean , the geometric mean , and the harmonic mean...

, together with the aforementioned arithmetic mean and the

harmonic meanIn mathematics, the harmonic mean is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired....

. 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 meansIn 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...

.)

## Calculation

The geometric mean of a data set

is given by:

The geometric mean of a data set

is less thanIn 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...

the data set's

arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

unless all members of the data set are equal, in which case the geometric and arithmetic means are equal. This allows the definition of the

arithmetic-geometric mean, a mixture of the two which always lies in between.

The geometric mean is also the

**arithmetic-harmonic mean** in the sense that if two

sequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...

s (

*a*_{n}) and (

*h*_{n}) are defined:

and

then

*a*_{n} and

*h*_{n} will converge to the geometric mean of

*x* and

*y*.

This can be seen easily from the fact that the sequences do converge to a common limit (which can be shown by

Bolzano–Weierstrass theoremIn real analysis, the Bolzano–Weierstrass theorem is a fundamental result about convergence in a finite-dimensional Euclidean space Rn. The theorem states thateach bounded sequence in Rn has a convergent subsequence...

) and the fact that geometric mean is preserved:

Replacing the arithmetic and harmonic mean by a pair of

generalized meanIn mathematics, a generalized mean, also known as power mean or Hölder mean , is an abstraction of the Pythagorean means including arithmetic, geometric, and harmonic means.-Definition:...

s of opposite, finite exponents yields the same result.

### Relationship with arithmetic mean of logarithms

By using

logarithmic identities- Trivial identities :Note that logb is undefined because there is no number x such that bx = 0. In fact, there is a vertical asymptote on the graph of logb at x = 0.- Canceling exponentials :...

to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication.

This is sometimes called the

**log-average**. It is simply computing the

arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

of the logarithm-transformed values of

(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

*f*(

*x*) = log

*x*. For example, the geometric mean of 2 and 8 can be calculated as:

where

*b* is any base of a

logarithmThe logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: More generally, if x = by, then y is the logarithm of x to base b, and is written...

(commonly 2,

*e*The mathematical constant ' is the unique real number such that the value of the derivative of the function at the point is equal to 1. The function so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base...

or 10).

### Relationship with arithmetic mean and mean-preserving spread

If a set of non-identical numbers is subjected to a

mean-preserving spreadIn probability and statistics, a mean-preserving spread is a change from one probability distribution A to another probability distribution B, where B is formed by spreading out one or more portions of A's probability density function while leaving the mean unchanged...

— that is, two or more elements of the set are "spread apart" from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.

### Computation in constant time

In cases where the geometric mean is being used to determine the average growth rate of some quantity, and the initial and final values

and

of that quantity are known, the product of the measured growth rate at every step need not be taken. Instead, the geometric mean is simply

where

is the number of steps from the initial to final state.

If the values are

, then the growth rate between measurement

and

is

. The geometric mean of these growth rates is just

## Properties

The fundamental property of the geometric mean, which can be proven to be false for any other mean, is

This makes the geometric mean the only correct mean when averaging

*normalized* results, that is results that are presented as ratios to reference values. 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 results:

| Computer A | Computer B | Computer C |

**Program 1** |
1 |
10 |
20 |

**Program 2** |
1000 |
100 |
20 |

**Arithmetic mean** |
500.5 |
55 |
**20** |

**Geometric mean** |
31.622... |
31.622... |
**20** |

The arithmetic and geometric means "agree" that computer C is the fastest. However, by presenting appropriately normalized values

*and* 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:

| Computer A | Computer B | Computer C |

**Program 1** |
1 |
10 |
20 |

**Program 2** |
1 |
0.1 |
0.02 |

**Arithmetic mean** |
**1** |
5.05 |
10.01 |

**Geometric mean** |
1 |
1 |
**0.632...** |

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

| Computer A | Computer B | Computer C |

**Program 1** |
0.1 |
1 |
2 |

**Program 2** |
10 |
1 |
0.2 |

**Arithmetic mean** |
5.05 |
**1** |
1.1 |

**Geometric mean** |
1 |
1 |
**0.632** |

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

### Proportional growth

The geometric mean is more appropriate than the

arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

for describing proportional growth, both

exponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...

(constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the

compound annual growth rateCompound annual growth rate is a business and investing specific term for the smoothed annualized gain of an investment over a given time period...

(CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount.

Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.6666% and 42.8571% for each year respectively. Using the

arithmetic meanIn mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

calculates a (linear) average growth of 46.5079% (80% + 16.6666% + 42.8579% divided by 3). However, if we start with 100 oranges and let it grow 46.5079% each year, the result is 314 oranges, not 300, so the linear average

*over*-states the year-on-year growth.

Instead, we can use the geometric mean. Growing with 80% corresponds to multiplying with 1.80, so we take the geometric mean of 1.80, 1.166666 and 1.428571, i.e.

; thus the "average" growth per year is 44.2249%. If we start with 100 oranges and let the number grow with 44.2249% each year, the result is 300 oranges.

### Applications in the social sciences

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 reduces 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.

Note that not all values used to compute the HDI are normalized; some of them instead have the form

. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

### Aspect ratios

The geometric mean has been used in choosing a compromise

aspect ratioThe aspect ratio of an image is the ratio of the width of the image to its height, expressed as two numbers separated by a colon. That is, for an x:y aspect ratio, no matter how big or small the image is, if the width is divided into x units of equal length and the height is measured using this...

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 aspect ratio their geometric mean.

In the choice of 16:9 aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is

, and thus 16:9 = 1.77... was chosen. This was discovered empirically 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 1.7:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.7:1. The value found by Powers is exactly the geometric mean of the extreme aspect ratios, 4:3 (1.33:1) and

CinemaScopeCinemaScope was an anamorphic lens series used for shooting wide screen movies from 1953 to 1967. Its creation in 1953, by the president of 20th Century-Fox, marked the beginning of the modern anamorphic format in both principal photography and movie projection.The anamorphic lenses theoretically...

(2.35:1), which is coincidentally close to 16:9 (1.78:1). Note that 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 the

14:914:9 is a compromise aspect ratio of 1.56:1. It is used to create an acceptable picture on both 4:3 and 16:9 televisions, conceived following audience tests conducted by the BBC...

(1.55...) aspect ratio, which is likewise used as a compromise between these ratios. In this case 14:9 is exactly the

*arithmetic mean*In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space...

of 16:9 and 4:3 = 12:9, since 14 is the average of 16 and 12, while the precise

*geometric mean* is

but the two different

*means*, arithmetic and geometric, are approximately equal because both numbers are sufficiently close to 1.

### Spectral flatness

In

signal processingSignal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

,

spectral flatnessSpectral flatness or tonality coefficient, also known as Wiener entropy, is a measure used in digital signal processing to characterize an audio spectrum. Spectral flatness, measured in decibels, provides a way to quantify how tone-like a sound is, as opposed to being noise-like...

, 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.

### Geometry

The length of the altitude of a

right triangleA right triangle or right-angled triangle is a triangle in which one angle is a right angle . The relation between the sides and angles of a right triangle is the basis for trigonometry.-Terminology:The side opposite the right angle is called the hypotenuse...

from the hypotenuse to the right angle, where the altitude is perpendicular to the hypotenuse, is the geometric mean of the two segments into which the hypotenuse is divided.

In an

ellipseIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...

, the

semi-minor axisIn geometry, the semi-minor axis is a line segment associated with most conic sections . One end of the segment is the center of the conic section, and it is at right angles with the semi-major axis...

is the geometric mean of the maximum and minimum distances of the ellipse from a focus; and the

semi-major axisThe major axis of an ellipse is its longest diameter, a line that runs through the centre and both foci, its ends being at the widest points of the shape...

of the ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.

## External links