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I came-up with a type of central tendency which is a weighted mean. The weighting is based on percentile, with values closer to the median having a higher weight. It's similar to the idea of a truncated mean, but it's a soft approach to dealing with outliers. For example, if x is the percentile of each value in the set, the weights can be determined with these 3 functions

x(x-1)
(x(x-1))²
(x(x-1))³

 Red=x(x-1)(4)  Blue=(x(x-1))²  Green=(x(x-1))³

I'm sure I'm not the first to come-up with this. Is there an existing term for this kind of central tendency?

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    $\begingroup$ Presumably you are using $x(1-x)$ or its powers $\endgroup$ – Henry Feb 13 '19 at 20:02
  • $\begingroup$ A normalized form for those weights is $x^n(1-x)^n(2n+1)!/(n!)^2$ $\endgroup$ – Matt F. Oct 6 '19 at 0:08
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Weightings based on percentiles also occur in rank-dependent expected utility (RDEU), so you could call these rank-dependent weightings or rank-dependent expectations.

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