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Harmonic Mean


The harmonic mean H(x_1,...,x_n) of n numbers x_i (where i=1, ..., n) is the number H defined by

 1/H=1/nsum_(i=1)^n1/(x_i).
(1)

The harmonic mean of a list of numbers may be computed in the Wolfram Language using HarmonicMean[list].

The special cases of n=2 and n=3 are therefore given by

H(x_1,x_2)=(2x_1x_2)/(x_1+x_2)
(2)
H(x_1,x_2,x_3)=(3x_1x_2x_3)/(x_1x_2+x_1x_3+x_2x_3),
(3)

and so on.

The harmonic means of the integers from 1 to n for n=1, 2, ... are 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, ... (OEIS A102928 and A001008).

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

 H=(G^2)/A
(4)

(Havil 2003, p. 120).

The harmonic mean is the special case M_(-1) of the power mean and is one of the Pythagorean means. In older literature, it is sometimes called the subcontrary mean.

The volume-to-surface area ratio for a cylindrical container with height h and radius r and the mean curvature of a general surface are related to the harmonic mean.

Hoehn and Niven (1985) show that

 H(a_1+c,a_2+c,...,a_n+c)>c+H(a_1,a_2,...,a_n)
(5)

for any positive constant c.


See also

Arithmetic Mean, Arithmetic-Geometric Mean, Geometric Mean, Harmonic-Geometric Mean, Harmonic Range, Power Mean, Pythagorean Means, Root-Mean-Square

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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. "Harmonic Mean." §4.13 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 57-58, 1962.Sloane, N. J. A. Sequences A001008/M2885 and A102928 in "The On-Line Encyclopedia of Integer Sequences."Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 602, 1995.

Referenced on Wolfram|Alpha

Harmonic Mean

Cite this as:

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

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