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 2`n` miles. The average speed (distance per
time) is thus:

2`n` / {(`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