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According to Wikipedia's example,

"the 20th percentile is the value (or score) below which 20% of the observations may be found"

If I have an inverse cumulative distribution function (instead of a normal one), does this means that e.g., P10 = a value that has 10% chance to be the P10 value or higher?

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    $\begingroup$ An inverse cumulative distribution function (aka quantile function) might be for a normal distribution. That aside, P10 would usually be the 10th percentile, so you just change the number in the definition "the 10th percentile is the value (or score) below which 10% of the observations may be found" (emphasis added). So your "higher" should be "lower". Values equal to any percentile require more detailed definitions, but on this evidence Wikipedia has the right flavour here. $\endgroup$ – Nick Cox Apr 13 '18 at 12:52
  • $\begingroup$ @Nick I suspect the use of "normal one" in the question might refer (somewhat confusingly) to "the cdf itself." I suspect (quite speculatively) that this question hinges on a misunderstanding of what "inverse" means in "inverse cdf." JrCaspian, could you clarify these issues for us? $\endgroup$ – whuber Apr 13 '18 at 14:48
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The inverse cummulative distribution function (or quantile function) is just, if it exists, the inverse of the commulative distribution function

In other words: Let $F$ be a strictly increasing and continuous cummulative distribution function in a domain $D \subset \mathbb{R}$. We define the inverse cummulative distribution function as $F^{-1}: [0,1] \to D$ in such a way that $F^{-1}(p)$ is the only real value $x \in D$ such that $F(x)=p$

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