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The harmonic mean of x1,...,xn is

n / (1/x1 + ... + 1/xn)
As the name implies, it's a mean (between the smallest and largest values).

An extension to the AM-GM inequality relates the harmonic mean to its more famous cousins, the geometric mean and the arithmetic mean.

The word "harmonic" is often used when inverses are involved; doubtless the harmonic mean has (Pythagorean, and therefore mystical) connections to musical harmony and pleasing intervals.

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

The harmonic mean also arises when a circuit contains multiple resistors in parellel. For example, with three parellel resistors of resistance A, B and C ohms respectively, the equivilant resistance of the unit is
R = 1 / (1/A + 1/B + 1/C)

Consequently, the equivilant resistance is always less than that of any of the individual resistors. The reason for the equation is obvious when you consider what portion of current goes through each resistor, and therefore what resistance is applied to each portion of the current.

Clearly, the harmonic mean is undefined if any of the elements are 0, because division by 0 is undefined. However, the integrity of the formula for resistance is maintained because nothing really has no resistance (outside the bizarre world of superconducters), although connecting wires themselves are usually treated as such.

Because the harmonic mean is lower than any of the individual elements, it is useful (theoretically) for giving false impressions about data. Although this would rarely occur in a real situation, I remember using it in a statistics homework about using averages to make data appear favourable :)

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