Geometric mean

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, an intersection of the two which always lies in between.

The geometric mean is also the arithmetic-harmonic mean in the sense that if two sequences (${\textstyle a_{n}}$) and (${\textstyle h_{n}}$) are defined:

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

and

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

where ${\textstyle h_{n+1}}$ is the harmonic mean of the previous values of the two sequences, then ${\textstyle a_{n}}$ and ${\textstyle h_{n}}$ will converge to the geometric mean of ${\textstyle x}$ and ${\textstyle y}$. The sequences converge to a common limit, and the geometric mean is preserved:

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

Replacing the arithmetic and harmonic mean by a pair of generalized means of opposite, finite exponents yields the same result.

The geometric mean can also be expressed as the exponential of the arithmetic mean of logarithms.[4] By using logarithmic identities to transform the formula, the multiplications can be expressed as a sum and the power as a multiplication:

When ${\displaystyle a_{1},a_{2},\dots ,a_{n}>0}$

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

As:

${\displaystyle {\begin{aligned}\left(\prod _{i=1}^{n}a_{i}\right)^{\frac {1}{n}}&={\sqrt[{n}]{a_{1}a_{2}\cdots a_{n}}}\\&=e^{\ln(a_{1}a_{2}\cdots a_{n})^{1/n}}\\&=e^{{\frac {1}{n}}\left(\ln a_{1}+\ln a_{2}+\cdots +\ln a_{n}\right)}\\&=e^{{\frac {1}{n}}\sum _{i=1}^{n}\ln a_{i}}\\{\text{geometric mean(}}a{\text{)}}&=e^{{\text{arithmetic mean(ln(}}a{\text{))}}}\end{aligned}}}$

alternatively, use any positive real number base, for both the logarithms and the number you are raising to the power of the arithmetic mean of the individual logarithms at that same base.

additionally, if negative values of the ${\displaystyle a_{i}}$ are allowed,

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

where m is the number of negative numbers.

This is sometimes called the log-average (not to be confused with the logarithmic average). It is simply computing the arithmetic mean of the logarithm-transformed values of ${\displaystyle a_{i}}$ (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 ${\displaystyle f(x)=\log x}$. For example, the geometric mean of 2 and 8 can be calculated as the following, where ${\displaystyle b}$ is any base of a logarithm (commonly 2, ${\displaystyle e}$ or 10):

${\displaystyle b^{{\frac {1}{2}}\left[\log _{b}(2)+\log _{b}(8)\right]}=4}$

Related to the above, it can be seen that for a given sample of points ${\displaystyle a_{1},\ldots ,a_{n}}$, the geometric mean is the minimizer of

${\displaystyle f(a)=\sum _{i=1}^{n}(\log(a_{i})-\log(a))^{2}=\sum _{i=1}^{n}(\log(a_{i}/a))^{2}}$,

whereas the arithmetic mean is the minimizer of

${\displaystyle f(a)=\sum _{i=1}^{n}(a_{i}-a)^{2}}$.

Thus, the geometric mean provides a summary of the samples whose exponent best matches the exponents of the samples (in the least squares sense).

The log form of the geometric mean is generally the preferred alternative for implementation in computer languages because calculating the product of many numbers can lead to an arithmetic overflow or arithmetic underflow. This is less likely to occur with the sum of the logarithms for each number.

Proof without words of the inequality of arithmetic and geometric means:
${\displaystyle PR}$ is the diameter of a circle centered on ${\displaystyle O}$; its radius ${\displaystyle AO}$ is the arithmetic mean of ${\displaystyle a}$ and ${\displaystyle b}$. Using the geometric mean theorem, triangle ${\displaystyle PGR}$'s altitude ${\displaystyle GQ}$ is the geometric mean. For any ratio ${\displaystyle a:b}$, ${\displaystyle AO\geq GQ}$.

Geometric proof without words that max (a,b) > root mean square (RMS) or quadratic mean (QM) > arithmetic mean (AM) > geometric mean (GM) > harmonic mean (HM) > min (a,b) of two distinct positive numbers a and b [note 1]

The geometric mean of a non-empty data set of (positive) numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 2 and 3 is 2.45, while their arithmetic mean is 2.5. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.[5]

In many cases the geometric mean is the best measure to determine the average growth rate of some quantity. (For example, if in one year sales increases by 80% and the next year by 25%, the end result is the same as that of a constant growth rate of 50%, since the geometric mean of 1.80 and 1.25 is 1.50.) In order to determine the average growth rate, it is not necessary to take the product of the measured growth rates at every step. Let the quantity be given as the sequence ${\displaystyle a_{0},a_{1},...,a_{n}}$, where ${\displaystyle n}$ is the number of steps from the initial to final state. The growth rate between successive measurements ${\displaystyle a_{k}}$ and ${\displaystyle a_{k+1}}$ is ${\displaystyle a_{k+1}/a_{k}}$. The geometric mean of these growth rates is then just:

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

The geometric mean is more appropriate than the arithmetic mean 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% + 42.8571%, that sum then 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. ${\displaystyle {\sqrt[{3}]{1.80\times 1.166666\times 1.428571}}\approx 1.442249}$; 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.

The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean.[8] It is also used in the CPI calculation[9] and recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union.

This has the effect of understating movements in the index compared to using the arithmetic mean.[8]

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 decreases 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.[10]

Not all values used to compute the HDI (Human Development Index) are normalized; some of them instead have the form ${\displaystyle \left(X-X_{\text{min}}\right)/\left(X_{\text{norm}}-X_{\text{min}}\right)}$. This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above.

The equally distributed welfare equivalent income associated with an Atkinson Index with an inequality aversion parameter of 1.0 is simply the geometric mean of incomes. For values other than one, the equivalent value is an Lp norm divided by the number of elements, with p equal to one minus the inequality aversion parameter.

The altitude of a right triangle from its right angle to its hypotenuse is the geometric mean of the lengths of the segments the hypotenuse is split into. Using Pythagoras' theorem on the 3 triangles of sides (p + q, r, s ), (r, p, h ) and (s, h, q ),
${\displaystyle {\begin{aligned}(p+q)^{2}\;\;&=\quad r^{2}\;\;\,+\quad s^{2}\\p^{2}\!\!+\!2pq\!+\!q^{2}&=\overbrace {p^{2}\!\!+\!h^{2}} +\overbrace {h^{2}\!\!+\!q^{2}} \\2pq\quad \;\;\;&=2h^{2}\;\therefore h\!=\!{\sqrt {pq}}\\\end{aligned}}}$

In the case of a right triangle, its altitude is the length of a line extending perpendicularly from the hypotenuse to its 90° vertex. Imagining that this line splits the hypotenuse into two segments, the geometric mean of these segment lengths is the length of the altitude. This property is known as the geometric mean theorem.

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-major axis of an ellipse is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix.

Another way to think about it is as follows:

Consider a circle with radius ${\displaystyle r}$. Now take two diametrically opposite points on the circle and apply pressure from both ends to deform it into an ellipse with semi-major and semi-minor axes of lengths ${\displaystyle a}$ and ${\displaystyle b}$.

Since the area of the circle and the ellipse stays the same, we have:

${\displaystyle {\begin{aligned}\pi r^{2}&=\pi ab\\r^{2}&=ab\\r&={\sqrt {ab}}\end{aligned}}}$

The radius of the circle is the geometric mean of the semi-major and the semi-minor axes of the ellipse formed by deforming the circle.

Distance to the horizon of a sphere (ignoring the effect of atmospheric refraction when atmosphere is present) is equal to the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere.

The geometric mean is used in both in the approximation of squaring the circle by S.A. Ramanujan[11] and in the construction of the heptadecagon with "mean proportionals".[12]

Equal area comparison of the aspect ratios used by Kerns Powers to derive the SMPTE 16:9 standard.[13]   TV 4:3/1.33 in red,   1.66 in orange,   16:9/1.77 in blue,   1.85 in yellow,   Panavision/2.2 in mauve and   CinemaScope/2.35 in purple.

The geometric mean has been used in choosing a compromise aspect ratio 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 the aspect ratio of their geometric mean.

In the choice of 16:9 aspect ratio by the SMPTE, balancing 2.35 and 4:3, the geometric mean is ${\textstyle {\sqrt {2.35\times {\frac {4}{3}}}}\approx 1.7701}$, and thus ${\textstyle 16:9=1.77{\overline {7}}}$... 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 1.77:1 and all of them also covered a smaller common inner rectangle with the same aspect ratio 1.77:1.[13] 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 ${\textstyle 16:9}$ (${\textstyle 1.77{\overline {7}}:1}$). 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 (${\textstyle 1.55{\overline {5}}}$...) aspect ratio, which is likewise used as a compromise between these ratios.[14] In this case 14:9 is exactly the arithmetic mean of ${\textstyle 16:9}$ and ${\textstyle 4:3=12:9}$, since 14 is the average of 16 and 12, while the precise geometric mean is ${\textstyle {\sqrt {{\frac {16}{9}}\times {\frac {4}{3}}}}\approx 1.5396\approx 13.8:9,}$ but the two different means, arithmetic and geometric, are approximately equal because both numbers are sufficiently close to each other (a difference of less than 2%).

The geometric mean is also used to calculate B and C series paper formats. The ${\displaystyle B_{n}}$ format has an area which is the geometric mean of the areas of ${\displaystyle A_{n}}$ and ${\displaystyle A_{n-1}}$. For example, the area of a B1 paper is ${\displaystyle {\frac {\sqrt {2}}{2}}\mathrm {m} ^{2}}$, because it is the geometric mean of the areas of an A0 (${\displaystyle 1\mathrm {m} ^{2}}$) and an A1 (${\displaystyle {\frac {1}{2}}\mathrm {m} ^{2}}$) paper (${\displaystyle {\sqrt {1\mathrm {m} ^{2}\cdot {\frac {1}{2}}\mathrm {m} ^{2}}}={\sqrt {{\frac {1}{2}}\mathrm {m} ^{4}}}={\frac {1}{\sqrt {2}}}\mathrm {m} ^{2}={\frac {\sqrt {2}}{2}}\mathrm {m} ^{2}}$).

The same principle applies with the C series, whose area is the geometric mean of the A and B series. For example, the C4 format has an area which is the geometric mean of the areas of A4 and B4.

An advantage that comes from this relationship is that an A4 paper fits inside a C4 envelope, and both fit inside a B4 envelope.

• 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.
• Anti-reflective coatings: In optical coatings, where reflection needs to be minimised between two media of refractive indices n₀ and n₂, the optimum refractive index n₁ of the anti-reflective coating is given by the geometric mean: ${\displaystyle n_{1}={\sqrt {n_{0}n_{2}}}}$.
• Subtractive color mixing: The spectral reflectance curve for paint mixtures (of equal tinting strength, opacity and dilution) is approximately the geometric mean of the paints' individual reflectance curves computed at each wavelength of their spectra.[15]
• Image processing: The geometric mean filter is used as a noise filter in image processing.
• Labor compensation: The geometric mean of a subsistence wage and market value of the labor using capital of employer was suggested as the natural wage by Johann von Thünen in 1875.[16]