Geometric Mean

The geometric mean of a sequence  is defined by

(1)

Thus,

			(2)
			(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 , 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 .

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.