The geometric mean of a sequence 
is defined by
 |
(1) |
Thus,
and so on.
The geometric mean of a list of numbers may be computed using GeometricMean[list] in the Wolfram Language package DescriptiveStatistics` .
For 
,
the geometric mean is related to the arithmetic mean 
and harmonic
mean 
by
 |
(4) |
(Havil 2003, p. 120).
The geometric mean is the special case 
of the power mean and is one
of the Pythagorean means.
Hoehn and Niven (1985) show that
 |
(5) |
for any positive constant 
.
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
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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." ยง4.10 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
Cite this as:
Weisstein, Eric W. "Geometric Mean." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeometricMean.html