Geometric mean
Encyclopedia
The geometric mean, in mathematics
, is a type of mean
or average
, which indicates the central tendency or typical value of a set of numbers. It is similar to the arithmetic mean
, except that the numbers are multiplied and then the nth root
(where n is the count of numbers in the set) of the resulting product
is taken.
For instance, the geometric mean of two numbers, say 2 and 8, is just the square root
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 geometry
. The geometric mean of two numbers, a and b, is the length of one side of a square
whose area is equal to the area of a 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 cube
whose volume is the same as that of a cuboid
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 population
or interest rates of a financial investment.
The geometric mean is also one of the three classic Pythagorean means
, together with the aforementioned arithmetic mean and the harmonic 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 (see Inequality of arithmetic and geometric means
.)
is given by:
The geometric mean of a data set is less than
the data set's arithmetic 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 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 sequence
s (an) and (hn) are defined:
and
then an and hn 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 theorem
) and the fact that geometric mean is preserved:
Replacing the arithmetic and harmonic mean by a pair of generalized mean
s of opposite, finite exponents yields the same result.
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 mean
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 logarithm
(commonly 2, e
or 10).
— 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.
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
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:
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:
while normalizing by B's result gives B as the fastest computer according to the arithmetic mean:
In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.
for describing proportional growth, both exponential growth
(constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate
(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 mean
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.
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.
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 CinemaScope
(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 14:9
(1.55...) aspect ratio, which is likewise used as a compromise between these ratios. In this case 14:9 is exactly the arithmetic mean
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
, 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.
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 ellipse
, the semi-minor axis
is the geometric mean of the maximum and minimum distances of the ellipse from a focus; and the semi-major axis
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.
Mathematics
Mathematics 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 mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
or average
Average
In 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 mean
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...
, except that the numbers are multiplied and then the nth root
Nth root
In 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 product
Product (mathematics)
In 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 root
Square root
In 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 satisfiesand 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 geometry
Geometry
Geometry 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 square
Square (geometry)
In 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 rectangle
Rectangle
In 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 cube
Cube
In 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 cuboid
Cuboid
In 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 population
World population
The 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 means
Pythagorean means
In 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 mean
Harmonic mean
In 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 means
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...
.)
Calculation
The geometric mean of a data set is given by:The geometric mean of a data set is less than
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...
the data set's arithmetic mean
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...
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 sequence
Sequence
In 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 (an) and (hn) are defined:
and
then an and hn 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 theorem
Bolzano–Weierstrass theorem
In 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 mean
Generalized mean
In 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 identitiesLogarithmic 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 mean
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 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 logarithm
Logarithm
The 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
E (mathematical constant)
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 spreadMean-preserving spread
In 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 simplywhere
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 justProperties
The fundamental property of the geometric mean, which can be proven to be false for any other mean, isThis 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 meanArithmetic 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...
for describing proportional growth, both exponential growth
Exponential growth
Exponential 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 rate
Compound annual growth rate
Compound 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 mean
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...
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 ratioAspect ratio (image)
The 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
CinemaScope 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 14:9
14:9
14:9 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
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
Signal 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 flatness
Spectral flatness
Spectral 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 triangleRight triangle
A 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 ellipse
Ellipse
In 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 axis
Semi-minor axis
In 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 axis
Semi-major axis
The 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
- 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