contraharmonic mean

English

Etymology

The contraharmonic mean of two variables is at the same distance from their arithmetic mean as their harmonic mean but on the opposite side.

Noun

contraharmonic mean (plural contraharmonic means)

1. (mathematics) A type of average calculated as the arithmetic mean of the squares of the values divided by the arithmetic mean of the values, ie. ${\displaystyle C={{(x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}) \over n} \over {(x_{1}+x_{2}+...+x_{n}) \over n}}\ or\ C(x_{1},x_{2},...,x_{n})={{x_{1}^{2}+x_{2}^{2}+...+x_{n}^{2}} \over {x_{1}+x_{2}+...+x_{n}}}}$

   The contraharmonic mean of two numbers ${\displaystyle a}$ and ${\displaystyle c}$ is the number ${\displaystyle b}$ such that ${\displaystyle {a-b \over c}={b-c \over a}}$. Combining the two ratios: ${\displaystyle {b-c \over a}={(a-b)+(b-c) \over c+a}={a-c \over a+c}}$. Rotating the proportion 90° clockwise: ${\displaystyle {a \over a+c}={b-c \over a-c}}$; transposing twice to isolate the ${\displaystyle b}$: ${\displaystyle b=c+a\left({a-c \over a+c}\right)}$. Likewise ${\displaystyle b=a-c\left({a-c \over a+c}\right)}$; these are both equivalent to the more symmetrical ${\displaystyle b={a^{2}+c^{2} \over a+c}}$.

Translations

See also

• arithmetic mean
• centroidal mean
• harmonic mean
• heronian mean
• geometric mean
• quadratic mean

References

• Means, Meaning, and Music: Pythagoras, Archytas, and Plato by Scott Makeig, §The Tetractus of Means