The harmonic mean is a useful
measure of central tendency for data
that consists of
rates or
frequencies. The concept
was named by
Archytas of Tarentum (
ca 428 BC,
Tarentum -
ca 350 BC,
Magna Graecia), a well known
mathematician,
statesman and
philosopher of the
Pythagorean School. In earlier
times, the harmonic mean was called the
sub-contrary mean but Archytas renamed it
harmonic since the ratio proved to be useful for
generating harmonious frequencies on string instruments.
Archytas was working on the "doubling of the cube" problem (the
Delian Problem); that is to find the side of a cube with a volume
twice that of a given cube. This problem had been worked on by
Hippocrates, but Archytas derived an elegant geometric solution using the harmonic mean.
A description of the harmonic mean is given by Plato (who was a close friend of Archytas):
One exceeding one extreme and being exceeded by the other by
the same fraction of the extremes.- Plato, Timaeus
In formula:
(c-a)/a = (b-c)/b
where a is the smallest term, b is the largest term, and c is the
middle term.
Rewritten for c, as the harmonic mean of a and b:
c = 2ab/(a+b)
1/c = 1/2 * {(1/a) + (1/b)}
The last equation can easily be rewritten to the extended form given by
ariels.
An example of the use of the harmonic mean: Suppose we're driving a
car from Amherst (A) to Boston (B) at a constant speed of 60 miles
per hour. On the way back from B to A, we drive a constant speed of 30
miles per hour (damn Turnpike). What is the average speed for the
round trip?
We would be inclined to use the arithmetic mean; (60+30)/2 = 45 miles
per hour. However, this is incorrect, since we have driven for a
longer time on the return leg. Let's assume the distance between A and B
is n miles. The first leg will take us n/60 hours,
and the return leg will take us n/30 hours. Thus, the total
round trip will take us (n/60) + (n/30) hours to
cover a distance of 2n miles. The average speed (distance per
time) is thus:
2n / {(n/60) +
(n/30)} = 2 / (1/20) = 40 miles per hour.
The reason that the harmonic mean is the correct average here is
that the numerators of the original ratios to be averaged
were equal (i.e. n miles at 60 miles/hour versus
n miles at 30 miles/hour). In cases where the
denominators of two ratios are averaged, we can use the
arithmetic mean.
factual sources:
http://mathforum.org/dr.math/problems/hasul12.15.96.html
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Archytas.html
William S. Peters, Counting for Something - Statistical Principles and Personalities, Springer Verlag, 1986