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I met the following equation.

$b=\frac{\sum_{i=1}^{N}x_i^2 }{\sum_{i=1}^{N}x_i }$, $0<x_i<1$

$x_i$ are probabilities.

Is it a kind of mean value? It seems that it is used as a mean value related to the fluctuation degree of a series of data. Is it right?

If so, in what case should it be used? And what is the difference between it and arithmetic mean or geometric mean?

if not, what does the equation mean?

update

In wiki I found that the equation above is contra-harmonic mean.

(link: https://en.wikipedia.org/wiki/Contraharmonic_mean )

I'm still wondering in what case should contra-harmonic mean be used. For example, when calculating the average of growth rates, geometric mean should be choosen. But what about contraharmonic mean?

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    $\begingroup$ Could you tell us in which context you did encounter this formula? without context it is pretty nlittle we can say. $\endgroup$
    – kjetil b halvorsen ♦
    Oct 27, 2015 at 16:18
  • $\begingroup$ @kjetilbhalvorsen I'm sorry that I didn't explain clearly. $x_i$ is from a discrete probability distribution. I think $b$ in the equation above is used as a approximation of the probability in a region. But I don't know why such approximation is used, rather than arithmetic mean or geometric mean, etc. $\endgroup$
    – fisher
    Oct 28, 2015 at 1:50
  • $\begingroup$ You should add new information as edits to the original post, and not as comments. And we still could use still more context! $\endgroup$
    – kjetil b halvorsen ♦
    Oct 28, 2015 at 8:16

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Are you sure that the xi's are probabilities? Why isn't the denominator one? This looks like the ratio of the first two noncentral sample moments. It appears to satisfy the conditions for a mean (symmetric, monotone increasing, M(ax)=aM(x), continuous, if a<=b then a<=M(a,b)<=b, etc).

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    $\begingroup$ Welcome to the site, @RichKenefic. This is really more of a comment than an answer. Note that CV is not a discussion forum, & that you will be able to comment anywhere when your reputation is >50. $\endgroup$
    – gung - Reinstate Monica
    Oct 27, 2015 at 18:12
  • $\begingroup$ @RichKenefic Thanks for your answer. $x_i$ is from a discrete probability distribution. And $b$ seems a approximation of $x_i$ within a region. But I don't know why such approximation is used, rather than arithmetic mean or geometric mean, etc $\endgroup$
    – fisher
    Oct 28, 2015 at 1:58

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