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
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
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