TOPICS
Harmonic Mean
The harmonic mean
of
numbers
(where
, ...,
) is the number
defined by
 | (1) |
The harmonic mean of a list of numbers may be computed in theWolframLanguage usingHarmonicMean[list].
The special cases of
and
are therefore given by
 |  |  | (2) |
 |  |  | (3) |
and so on.
The harmonic means of the integers from 1 to
for
, 2, ... are 1, 4/3, 18/11, 48/25, 300/137, 120/49, 980/363, ... (OEISA102928 andA001008).
For
, the harmonic mean is related to thearithmetic mean
andgeometric mean
by
 | (4) |
(Havil 2003, p. 120).
The harmonic mean is the special case
of thepower mean and is one of thePythagorean means. In older literature, it is sometimes called the subcontrary mean.
Thevolume-to-surface area ratio for a cylindrical container with height
and radius
and themean curvature of a general surface are related to the harmonic mean.
Hoehn and Niven (1985) show that
 | (5) |
for anypositive constant
.
See also
Arithmetic Mean,Arithmetic-Geometric Mean,Geometric Mean,Harmonic-Geometric Mean,Harmonic Range,Power Mean,Pythagorean Means,Root-Mean-SquareExplore 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. "Harmonic Mean." §4.13 inMathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 57-58, 1962.Sloane, N. J. A. SequencesA001008/M2885 andA102928 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 MeanCite this as:
Weisstein, Eric W. "Harmonic Mean." FromMathWorld--A Wolfram Web Resource.https://mathworld.wolfram.com/HarmonicMean.html