Contraharmonic mean describes a mean of a set of numbers that is complementary to the harmonic mean. In mathematics, the harmonic mean is one of several methods of calculating an average. ...

Summary

The contraharmonic mean of a set of positive numbers is defined as the Arithmetic mean of the squares of the numbers divided by the Arithmetic mean of the numbers: In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

or, more simply,

It is easy to show that this satisfies the characteristic properties of a mean: In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ...

The first property implies that for all k > 0,

C(k, k, ..., k) = k (fixed point property).

For two variables, a and b, taken as

it is easier to see why this mean is complementary to the harmonic mean. Then the contraharmonic mean C(a,b) is also that mean that is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean: In mathematics, the harmonic mean is one of several methods of calculating an average. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

C(a,b) - A(a,b) = A(a,b) - H(a,b)

or

C(a,b) = 2A(a,b) - H(a,b).

Explication

From the formulas for the arithmetic mean and harmonic mean of two variables we have :

and

=

Notice that for two variables the Average of the Harmonic and Contraharmonic means is exactly equal to the Arithmetic mean:

A( H(a,b), C(a,b) ) = A(a,b)

As a gets closer to 0 then H(a,b) also gets closer to 0. The harmonic mean is very sensitive to low values. On the other hand, the contraharmonic mean is sensitive to larger values, so as a approaches 0 then C(a,b) approaches b (so their average remains A(a,b) ).

The contraharmonic mean is higher in value than the average and also higher than the root mean square : In mathematics, the root mean square or rms is a statistical measure of the magnitude of a varying quantity. ...

where H is the harmonic mean, G is geometric mean, L is the logarithmic mean, A is the arithmetic mean, R is the root mean square and C is the contraharmonic mean. If a could be equal to b then the above < signs can be replaced by . When a=b the above chain of comparisons collapses to the same value, a.
The geometric mean of a set of positive data is defined as the nth root of the product of all the members of the set, where n is the number of members. ... In mathematics, the logarithmic mean is a function of two numbers which is equal to their difference divided by the logarithm of their quotient. ...

There are two other notable relationships between 2-variable means. First, the geometric mean of the arithmetic and harmonic means is equal to the Geometric mean of the two values :

The second relationship is that the Geometric mean of the arithmetic and contraharmonic means is the root mean square:

 :

The contraharmonic mean of two variables can be constructed geometrically using a trapezoid (see [1] ).

Additional constructions

You can also construct it using a circle similar to the way the Pythagorean means of two variables are constructed.
Basically, the Contraharmonic mean is the rest of the diagonal on which the Harmonic mean lies. Their sum is always the length of the diagonal (a+b) of the circle that shows the Pythagorean means (see Pythagorean means on MathWorld at [2] ). The three classical Pythagorean means are the arithmetic mean (A), the geometric mean (G), and the harmonic mean (H). ...

References

• Essay #3 - Some "mean" Trapezoids, by Shannon Umberger: [3]

• Construction of the Contraharmonic Mean in a Trapezoid: [4]

• Means in the Trapezoid: [5]

• Means of Complex Numbers: [6]

• Proofs without Words / Exercises in Visual Thinking, by Roger B. Nelsen, page 56, ISBN 0-88385-700-6

• Pythagorean Means: [7] (extend the segment that represents the Harmonic mean through the circle's center to the other side, creating a diagonal. The length of the diagonal segment after the Harmonic segment is the Contraharmonic mean.)