Contraharmonic mean

Contraharmonic mean describes a mean of a set of numbers that is complementary to the harmonic mean. It is a special case of the Lehmer mean.

Definition
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:


 * $$C(x_1, x_2, ..., x_n) ={ { (x_1^2+x_2^2+...+x_n^2) \over n} \over {(x_1+x_2+...+x_n) \over n}}, $$

or, more simply,


 * $$C(x_1, x_2, ..., x_n) ={ { x_1^2+x_2^2+...+x_n^2} \over {x_1+x_2+...+x_n }}.$$

Properties
It is easy to show that this satisfies the characteristic properties of a mean:


 * $$ C(x_1, x_2, ..., x_n) \in [ \min(x_1, x_2, ..., x_n), \max(x_1, x_2, ..., x_n) ] $$
 * $$ C(t\cdot x_1, t\cdot x_2, ..., t\cdot x_n) = t\cdot C(x_1, x_2, ..., x_n)$$ for $$t > 0 $$

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


 * $$0 < a \le b ,$$

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:


 * 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 :


 * $$ A(a,b) = {{(a+b)} \over 2} $$ and
 * $$ H(a,b) = {1 \over {{1 \over 2}\cdot {({1 \over a} + {1 \over b})}}} $$ = $${2ab} \over {a+b}$$
 * $$ C(a,b) = 2\cdot A(a,b) - H(a,b) = (a+b)- {{2ab} \over {a+b}} $$
 * $$ C(a,b) = { {(a+b)^2 - 2ab } \over {a+b}} = {{a^2 + b^2} \over {a+b}} $$

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 :


 * $$ \min(a,b) < H(a,b) < G(a,b) < L(a,b) < A(a,b) < R(a,b) < C(a,b) < \max(a,b) \ for \ 0<a<b $$

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 $$ \le $$. When a=b the above chain of comparisons collapses to the same value, a.

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 :


 * $$ G( A(a,b), H(a,b) )=G\left({{a+b}\over 2}, {{2ab}\over {a+b}}\right) =

\sqrt {{{a+b}\over 2}\cdot {{2ab}\over {a+b}}} = \sqrt{ab} = G(a,b) $$

The second relationship is that the Geometric mean of the arithmetic and contraharmonic means is the root mean square:
 * $$ G( A(a,b), C(a,b) )=G\left({{a+b}\over 2}, {{a^2+b^2}\over {a+b}}\right) = $$ :$$\sqrt{{{a+b}\over 2}\cdot {{a^2+b^2}\over {a+b}}} = \sqrt = R(a,b) $$

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

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 ).