According to Sir Thomas Thomas Heath (1921, p. 85) in Pythagoras’s time, there were three means, the arithmetic, the geometric, and the subcontrary, and the “name of the third (‘subcontrary’) was changed by Archytas and Hippasus to harmonic.” In English, the term geometrical mean can be found as early as in 1695 in the E. Halley’s paper (Halley 1695–1697, p. 62). The geometric mean is a measure of central tendency that is “primarily employed within the context of certain types of analysis on business and economics” (Sheskin 2004, p. 8), such as an average of index numbers, ratios, and percent changes over time. For example, Fisher “ideal index” is the geometric mean of the Laspeyres index and the Paasche index. Geometric mean is also being used in modern portfolio theory and investment analysis (see, for example, Elton et al. 2009, p. 231), and in calculation of compound annual growth rate.
The geometric mean of n positive numbers x 1, x 2, . . . , x n is defined as positive nth root...
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References and Further Reading
Bullen PS (1987) Handbook of means and their inequalities, 2nd edn. Springer, Heidelberg
Elton EJ, Gruber MJ, Brown SJ, Goetzmann WN (2009) Modern portfolio theory and investment analysis, 8th edn. Wiley, New York
Halley E (1695–1697) A most compendious and facile method for constructing the logarithms, exemplified and demonstrated from the nature of numbers, without any regard to the hyperbola, with a speedy method for finding the number from the logarithm given. Phil Trans R Soc Lond 19:58–67
Heath T (1921) A history of Greek mathematics, vol. 1: from Thales to Euclid. Clarendon, Oxford
Sheskin DJ (2004) Handbook of parametric and nonparametric statistical procedures, 3rd edn. Chapman & Hall/CRC Press, Boca Raton
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Stević, S. (2011). Geometric Mean. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_644
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