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The contraharmonic mean of several positive numbers $u_{1}$, $u_{2}$, $\ldots$, $u_{n}$ is defined as

$c\;:=\;\frac{u_{1}^{2}\!+\!u_{2}^{2}\!+\ldots+\!u_{n}^{2}}{u_{1}\!+\!u_{2}\!+% \ldots+\!u_{n}}.$ |

This concept has certain applications; one of them is by [1] the following.

If $\langle u_{1}$, $u_{2}$, $\ldots$, $u_{n}\rangle$ is the distribution^{} of the seats of $n$ parties,
$s$ the total number of seats in the body of delegates
($u_{1}\!+\!u_{2}\!+\ldots+\!u_{n}\,=\,s$),
and one draws a random seat (with probability $1/s$), then the size of the drawn
delegate’s party has the expected value

$\frac{u_{1}}{s}\!\cdot\!u_{1}+\frac{u_{2}}{s}\!\cdot\!u_{2}+\ldots+\frac{u_{n}% }{s}\!\cdot\!u_{n}\;=\;c.$ |

# References

- 1
Caulier, Jean-François: The interpretation
^{}of the Laakso–Taagepera effective number of parties. – Documents de travail du Centre d’Economie de la Sorbonne (2011.06).

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## Mathematics Subject Classification

05A18*no label found*26E60

*no label found*60A05

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