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Geometric Mean -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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The geometric mean of a sequence {a_i}_(i=1)^n is defined by

G(a_1,...,a_n)=(product_(i=1)^na_i)^(1/n).

(1)

Thus,

and so on.

The geometric mean of a list of numbers may be computed using GeometricMean[list] in the Mathematica package DescriptiveStatistics` (which can be loaded with the command <<DescriptiveStatistics`) .

For n=2, the geometric mean is related to the arithmetic mean A and harmonic mean H by

G=sqrt(AH)

(4)

(Havil 2003, p. 120).

The geometric mean is the special case M_0 of the power mean and is one of the Pythagorean means.

Hoehn and Niven (1985) show that

G(a_1+c,a_2+c,...,a_n+c)>c+G(a_1,a_2,...,a_n)

(5)

for any positive constant c.

REFERENCES:

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 10, 1972.

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 119-121, 2003.

Hoehn, L. and Niven, I. "Averages on the Move." Math. Mag. 58, 151-156, 1985.

Kenney, J. F. and Keeping, E. S. "Geometric Mean." §4.10 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 54-55, 1962.

Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995.