TOPICS
Search

Geometric Mean


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,

G(a_1,a_2)=sqrt(a_1a_2)
(2)
G(a_1,a_2,a_3)=(a_1a_2a_3)^(1/3),
(3)

and so on.

The geometric mean of a list of numbers may be computed using GeometricMean[list] in the Wolfram Language package 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.


See also

Arithmetic Mean, Arithmetic-Geometric Mean, Arithmetic-Logarithmic-Geometric Mean Inequality, Carleman's Inequality, Harmonic Mean, Mean, Power Mean, Pythagorean Means, Root-Mean-Square Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

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.

Referenced on Wolfram|Alpha

Geometric Mean

Cite this as:

Weisstein, Eric W. "Geometric Mean." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeometricMean.html

Subject classifications