The harmonic mean
of numbers (where , ..., ) is the number defined by

(1)

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

The special cases of
and are therefore given by

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,
... (OEIS A102928 and A001008 ).

For ,
the harmonic mean is related to the arithmetic mean and geometric
mean
by

(4)

(Havil 2003, p. 120).

The harmonic mean is the special case 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 and radius and the mean curvature of
a general surface are related to the harmonic mean.

Hoehn and Niven (1985) show that

(5)

for any positive constant .

See also Arithmetic Mean ,

Arithmetic-Geometric Mean ,

Geometric Mean ,

Harmonic-Geometric
Mean ,

Harmonic Range ,

Power
Mean ,

Pythagorean Means ,

Root-Mean-Square
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.
"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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