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I'm trying to make the jump from the idea of a percentile, say, over the real number line (where the nth percentile is simply the position in which n% of data points are below it, and 100-n% are above it), to the idea of the area under a probability density function.

If I want to know the 50% percentile from a set of numbers, I'll find the point in which half the numbers are below, half the numbers are above. That's the 50% percentile, and I'm done.

If I want to know the 50% percentile from a distribution, say, a Z-score, I'll evaluate the cdf from 0 - 50, and I'm done. Am I saying this correct?

This feel right intuitively, but I need some discussion to hammer it home. Or, I could be completely off...

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You are close but not exactly right. Remember that the area under a probability distribution has to sum to 1. The cumulative density function (CDF) is a function with values in [0,1] since CDF is defined as $$ F(a) = \int_{-\infty}^{a} f(x) dx $$ where f(x) is the probability density function. Then 50th percentile is the total probability of 50% of the samples which means the point where CDF reaches 0.5. Or in more general terms, the p'th percentile is the point where the CDF reaches p/100.

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    $\begingroup$ Perhaps it's worth pointing out how close the OP got - instead of "evaluating a CDF" they should be evaluating an inverse CDF. $\endgroup$ – Silverfish Jan 11 '15 at 8:18
  • $\begingroup$ so close yet so far... :) $\endgroup$ – Matt O'Brien Dec 2 '15 at 20:40
  • $\begingroup$ In general, the inverse of a CDF (in the usual sense, i.e., inverse of a function) may not exist. We should consider the so-called generalized inverse (or pseudo-inverse) of a CDF. $\endgroup$ – Danny Pak-Keung Chan Sep 21 '17 at 21:46
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No. Essentially, calculating a percentile (or a p-quantile) is equivalent to finding the inverse of a CDF.

Note that the inverse, in the usual sense, of a CDF may not exist and the notion of generalized inverse should be introduced. To make discussion precise, we clarify all definitions.

Definition: A CDF is a function $F:[-\infty,\infty]\rightarrow[0,1]$ that satisfies the following conditions:

  1. (Increasing) For any $x,y\in[-\infty,\infty]$, if $x<y$, then $F(x)\leq F(y)$,

  2. (Right-continuity) For any $a\in\mathbb{R}$, we have that $F(a)=\lim_{x\rightarrow a+}F(x)$,

  3. $F(-\infty)=\lim_{x\rightarrow-\infty}F(x)=0$, and

  4. $F(\infty)=\lim_{x\rightarrow\infty}F(x)=1$.

We have at least two versions of generalized inverse of $F$, denoted by $Inv_{1}F$ and $Inv_{2}F$, which are defined as follows.

$Inv_{1}F:[0,1]\rightarrow[-\infty,\infty]$, defined by $Inv_{1}F(x)=\inf\{y\mid F(y)\geq x\},$

$Inv_{2}F:[0,1]\rightarrow[-\infty,\infty]$, defined by $Inv_{2}F(x)=\inf\{y\mid F(y)>x\}$.

Here, we adopt the convention that $\inf(\emptyset)=\infty$.

If I remember correctly, given $p\in[0,1]$, the $p$-quantile is simply defined as $Inv_{1}F(p)$.

Of course, if $F$ is strictly increasing and continuous, both versions of generalized inverse are the same and reduce to the usual inverse of function$F^{-1}:[0,1]\rightarrow[-\infty,\infty].$

For more information: https://people.math.ethz.ch/~embrecht/ftp/generalized_inverse.pdf

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