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 defined as the n th root of the product of n numbers, i. A geometric mean is often used when comparing different items—finding a single "figure of merit" for these items—when each item has multiple properties that have different numeric ranges.
Arithmetic vs. Geometric Return
If an arithmetic mean were used instead of a geometric mean, the financial viability would have greater weight because its numeric range is larger. That is, a small percentage change in the financial rating e. The use of a geometric mean normalizes the differently-ranged values, meaning a given percentage change in any of the properties has the same effect on the geometric mean.
The geometric mean can be understood in terms of geometry. The geometric mean applies only to positive numbers. The geometric mean is also one of the three classical 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.
The Difference Between Arithmetic Mean and Geometric Mean
The above figure uses capital pi notation to show a series of multiplications. Each side of the equal sign shows that a set of values is multiplied in succession the number of values is represented by "n" to give a total product of the set, and then the n th root of the total product is taken to give the geometric mean of the original set.
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. 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 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. This is sometimes called the log-average not to be confused with the logarithmic average. 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. If a set of non-identical numbers is subjected to a mean-preserving spread — 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.
The geometric mean of these growth rates is then just.
Geometric mean vs arithmetic mean pdf merge
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. 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 comparison of execution time of computer programs:.
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:. In all cases, the ranking given by the geometric mean stays the same as the one obtained with unnormalized values.
However, this reasoning has been questioned.
The Effect of Outliers
In general, it is more rigorous to assign weights to each of the programs, calculate the average weighted execution time using the arithmetic mean , and then normalize that result to one of the computers. The use of the geometric mean for aggregating performance numbers should be avoided if possible, because multiplying execution times has no physical meaning, in contrast to adding times as in the arithmetic mean.
Metrics that are inversely proportional to time speedup, IPC should be averaged using the harmonic mean. Similarly, this is possible for the weighted geometric mean. 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. Using the arithmetic mean calculates a linear average growth of However, if we start with oranges and let it grow Instead, we can use the geometric mean. If we start with oranges and let the number grow with 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. This has the effect of understating movements in the index compared to using the arithmetic mean. Although the geometric mean has been relatively rare in computing social statistics, starting from 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:.
This makes the choice of the geometric mean less obvious than one would expect from the "Properties" section above. 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. Distance to the horizon of a sphere is the geometric mean of the distance to the closest point of the sphere and the distance to the farthest point of the sphere. Both in the approximation of squaring the circle according to S.
Ramanujan and in the construction of the Heptadecagon according to "sent by T. Stowell, credited to Leybourn's Math. Repository, " , the geometric mean is employed. 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.
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. The intermediate ratios have no effect on the result, only the two extreme ratios.
Applying the same geometric mean technique to and approximately yields the 1. 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 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.
The geometric mean filter is used as a noise filter in image processing. From Wikipedia, the free encyclopedia.
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Unsourced material may be challenged and removed. Further information: Compound annual growth rate. Mathematics portal. Arithmetic-geometric mean Generalized mean Geometric mean theorem Geometric standard deviation Harmonic mean Heronian mean Hyperbolic coordinates Log-normal distribution Muirhead's inequality Multiplicative calculus Product Pythagorean means Quadratic mean Quadrature mathematics Rate of return Weighted geometric mean.
Retrieved 14 June David E. Joyce, Clark University. Retrieved 19 July Transaction Processing Performance Council. Archived from the original on 4 November Retrieved 9 January The definition is unambiguous if one allows 0 which yields a geometric mean of 0 , but may be excluded, as one frequently wishes to take the logarithm of geometric means to convert between multiplication and addition , and one cannot take the logarithm of 0. Statistics: An Introduction using R. The Mathematical Gazette.
Communications of the ACM. The Financial System Today.
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Manchester University Press. Archived from the original on The CinemaSource Press. Archived PDF from the original on Retrieved Outline Index. Descriptive statistics. Mean arithmetic geometric harmonic Median Mode. Central limit theorem Moments Skewness Kurtosis L-moments. Index of dispersion. Grouped data Frequency distribution Contingency table.
Formulas for Calculation
Data collection. Sampling stratified cluster Standard error Opinion poll Questionnaire. Scientific control Randomized experiment Randomized controlled trial Random assignment Blocking Interaction Factorial experiment.