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 the Wolfram Language using HarmonicMean[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, ... (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 
.
Contact the MathWorld Team
harmonic mean {1,2,3,4,5}
