Contraharmonic Mean - Two Variable Formulae

Two Variable Formulae

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

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) ).

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:

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

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