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I am really confused behind the mathematical meaning of a 100th percentile.

What does that mean mathematically? Does it mean that a data point in a sample space is greater in some metric and that is also greater than itself?

That makes no sense. AFAIK, there can be no such thing as the 100th percentile, because the maximum data point considered is still part of the sample space.

For example, when a student scores the highest marks, he/she can be the 99.999th percentile, but what is the meaning of 100th percentile ?

Mittens
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ng.newbie
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    Less mathematically inclined people tend to like to round numbers whenever possible. – JMoravitz Sep 16 '20 at 14:32
  • It is the maximum value that a specific random variable may take. In a sample, it is the maximum value of your sample. – Mittens Sep 16 '20 at 14:33
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    @OliverDiaz Yes but that cannot be 100th percentile. It can be something as 99.9999th percentile. Never 100th, close to 100th but never 100th. Right ? – ng.newbie Sep 16 '20 at 14:35
  • @JMoravitz So its just an approximation error ? – ng.newbie Sep 16 '20 at 14:35
  • Your remarks make sense to me. It is almost like asking "how severe does an earthquake have to be to register 10.0 on the richter scale", except that a (for example) meteor collision with earth that extinguishes $90^+$% of all human life (and is theoretically feasible) would probably trigger an earthquake that is assigned a 10.0 on the Richter. – user2661923 Sep 16 '20 at 14:36
  • @user2661923 Not sure about your example. But my question is 100th percentile a mathematically valid thing to say ? – ng.newbie Sep 16 '20 at 14:38
  • And my response is that I regard the assertion that it is invalid as reasonable. – user2661923 Sep 16 '20 at 14:40
  • @user2661923 But not entirely wrong ? – ng.newbie Sep 16 '20 at 14:41
  • I never studied statistics, so I am hamstrung. I will allow my intuition to assert that the idea that 100th percentile is invalid seems reasonable. However, since I never studied statistics, that's all that I can say. – user2661923 Sep 16 '20 at 14:43
  • When talking about describing sampled data, e.g. test scores, you might leave it open to the possibility that there might still exist someone who could have scored better and so you can not confidently say the largest score you saw is the largest score possible to see. For that reason, you might leave it at 99.9% or what have you. On the other hand if your goal is to talk only about those datapoints in the sample and not try to make conclusions about the population as a whole rather than only the sampled population, then calling it 100 for largest seen could make sense. – JMoravitz Sep 16 '20 at 14:46
  • The end result is that it depends on how you choose to define it or rather what definitions you choose to follow. And for that, there are in fact different options. As in many areas of math, there is often more than one way to define things. Most of these ways will be similar and fill similar purposes, but such as here have different nuances. – JMoravitz Sep 16 '20 at 14:48
  • @user2661923 The Richter scale is a logarithmic scale. A 10.0 would just mean an earthquake whose seismic amplitude is ten times as large as a 9.0. There is no upper bound and it is not analogous to percentiles in any way. A large asteroid collision would probably trigger a 11.5 or larger. – shalop Sep 16 '20 at 16:03
  • @shalop nice rebuttal, my mistake – user2661923 Sep 16 '20 at 16:05
  • Percentiles can sometimes be odd. For example this Wikipedia article gives at least 9 methods of determining quantiles from a sample https://en.wikipedia.org/wiki/Quantile#Estimating_quantiles_from_a_sample – Ralph Winters Apr 29 '22 at 18:26

2 Answers2

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I cannot comment there. So I am posting an answer.

When the sample space is infinitely large as compared to the point/interval you are referring to, then 100 percentile makes sense.

For instance, the value of $y(0)$ in the function $e^{-x^2}$ is a 100 percentile.

Natasha
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    what is $y(0)$? what is the (mathematical) definition of percentile (aka quantile) you are using? I don't quite follows the last line of your explanation. could you please elaborate? – Mittens Sep 16 '20 at 16:42
  • @Natasha: Your statement does not make sense. What is your definition of percentile, and what do you mean by "... the point you are referring to..."? – Mittens Sep 16 '20 at 18:45
  • Percentile is the percentage of inputs having outputs less than a particular output. As in his case, a student can never score 100 percentile as the total number of students below him must be less than 100% – Natasha Sep 17 '20 at 02:02
  • The value of the function at all other inputs is less than that at $0$. So percentage of inputs that are less that $y(0)$ is 100% in this case as the sample space is infinitely large. Hence 100 percentile makes sense here. – Natasha Sep 17 '20 at 02:04
  • That is not the mathematical definition of percentile. If X has uniform distribution in [0,1], what is the 0, 10%, 99%?and 100% percentile according to your notion of percentile. – Mittens Sep 17 '20 at 15:10
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Before attempting to give an answer to the OP, I briefly discuss the definition of a quantile (percentile is also used)

Definition:

  • Given a (cumulative) distribution $F$ on $\mathbb{R}$, for $0<q<1$ a $q$-quantile (or percentile) is a number $z_q$ such that $$ \begin{align} F(z_q-)\leq q\leq F(z_q)\tag{0}\label{quantile} \end{align} $$
  • Similarly, if $X$ is a real valued random variable with distribution $F_X$, for $0<q<1$ a $q$-quantile is a number $z_q$ such that $$ \begin{align}\mathbb{P}[X<z_q]=F_X(z_q-)\leq q\leq F(z_q)=\mathbb{P}[X\leq q]\tag{1}\label{quantileX} \end{align} $$

Some special quantile functions are

  • Consider the functions $Q$, $Q^+$ defined on $(0,1)$ by $$ \begin{align} Q(q)&=\inf\{x\in\mathbb{R}: F(x)\geq q\}\\ Q^+(q)&=\sup\{x\in\mathbb{R}:F(x-)\leq q\} \end{align} $$ It is not difficult to check that $Q(q)$ and $Q^+(q)$ satisfy \eqref{quantile} for each $0<q<1$. $Q$ and $Q_+$ have the additional property that for any other $q$-quantile $z_q$ of $F$ satisfies $$Q(q)\leq z_q\leq Q^+(q)$$

Back to the OP:

From the definition of a quantile, it is easy to see that

  1. Unless $F$ (or rather the Probability measure induced by $F$) is supported an a interval bounded from below, the definition of $q$-quantile cannot be expended to $q=0$.
  2. Unless $F$ is supported on an interval bounded from above, the definition of $q$-quantile cannot be extended to $q=1$.

However

  1. If $F$ is supported in an interval bounded from below, one can extend the notion of $q$-quatile with $q=0$ can be done, and the functions $Q$ and $Q^+$ can de defined at $q=0$ as extended real numbers $Q(0)=-\infty<Q^+(0)=\sup\{x:F(x-)=0\}$

  2. If $F$ is supported in an interval bounded from above, one can extend the notion of $q$-quatile with $q=1$, and the functions $Q$ and $Q^+$ can de defined at $q=1$ as extended real numbers $Q(1)=\in\{x:F(x-)=1\}<\infty=Q^+(1)$

For sample of size $n$ of some distribution $F$ on the real line, say $x_1,\ldots, x_n$ one typically considers the empirical distribution $$F_n(x)=\frac{1}{n}\sum^n_{k=1}\mathbb{1}_{(-\infty,x]}(x_k)$$ Of course $F_n$ is supported in the compact interval interval $[\min_kx_k,\max_kx_k]$, so there $0$ and $1$ quantiles are mathematically well deinided in this situation. There is no harm to define the $0$-quantile and the $1$-quantile of $F_n$ (or the sample X) as $\min_kx_k$ and $\max_kx_m$ respectively. In fact, many statistical packages display the $0$ and $1$ quantiles as the min and the max of the sample.

Mittens
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    If the support of a distribution $F$ is not bounded from below, $Q(0)$ and $Q^+(0)$ are still well defined as extended real numbers, $Q(0)=-\infty$ since for any number $x$: $Q(x)>0$, and $Q^+(0)=-\infty$ since the supreme of the empty set (logical extension) is $-\infty$. Similarly, if the support of $F$ is not bounded from above, $Q(1)$ and $Q^+(1)$ exist as extended real numbers and both are $+\infty$. – Mittens Sep 16 '20 at 16:11
  • @JeanLeider: but of course! The $0$-qiantiles being a vacuous set in the former case and the $1$-quantiles being a vacuous set in the later. – Mittens Sep 16 '20 at 16:16