Contraharmonic mean
In mathematics, a contraharmonic mean is a function complementary to the harmonic mean. The contraharmonic mean is a special case of the Lehmer mean, , where p=2.
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:
or, more simply,
It is easy to show that this satisfies the characteristic properties of a mean:
The first property implies that for all k > 0,
The contraharmonic mean is higher in value than the arithmetic mean and also higher than the root mean square:
where x is a list of values, 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. Unless all values of x are the same, the ≤ signs above can be replaced by <.
The name "contraharmonic" may be due to the fact that when taking the mean of only two variables, the contraharmonic mean is as high above the arithmetic mean as the arithmetic mean is above the harmonic mean (i.e., the arithmetic mean of the two variables is equal to the arithmetic mean of their harmonic and contraharmonic means).
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:
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