What is the difference between the three terms below?

  • percentile
  • quantile
  • quartile
  • 8
    $\begingroup$ A deeper question is whether quantiles etc. are intervals or points. $\endgroup$
    – Henry
    Jun 13 '15 at 14:20
  • 9
    $\begingroup$ The quantiles are defined as points. There is often ambiguity as between intervals and points for quartiles etc.; it does not bite very hard in practice, as context usually makes clear what is intended. I prefer the first quarter (rather than quartile), for the lowest 25%, etc. although it's too much to hope that the distinction will be universally self-evident without explanation. $\endgroup$
    – Nick Cox
    Jun 13 '15 at 17:22
  • 1
    $\begingroup$ My answer at stats.stackexchange.com/questions/235330/… has a fuller list of *ile terms, including dates of first use. Naturally additions and earlier sightings (citings!) are welcome. $\endgroup$
    – Nick Cox
    Jan 16 '19 at 16:43
  • $\begingroup$ Quartile relates to quarters, i.e. out of 4. Pencentile relates to percentages, i.e. out of 100. Quantile ... is just there to confuse you (it relates to quantity). $\endgroup$ Nov 15 '20 at 6:19

0 quartile = 0 quantile = 0 percentile

1 quartile = 0.25 quantile = 25 percentile

2 quartile = .5 quantile = 50 percentile (median)

3 quartile = .75 quantile = 75 percentile

4 quartile = 1 quantile = 100 percentile

  • 7
    $\begingroup$ In case anyone else was confused looking at this: this is not saying that a quantile varies between 0 and 1, and percentile between 0 and 100, it's saying that these are the domains of the quantile(x) and percentile(x) functions, which return an observed value, the range of which is completely dependent on your specific problem (e.g. if you are measuring rainfall it's probably between 0 and 10). $\endgroup$ Apr 18 '19 at 22:01
  • Percentiles go from $0$ to $100$.

  • Quartiles go from $1$ to $4$ (or $0$ to $4$).

  • Quantiles can go from anything to anything.

  • Percentiles and quartiles are examples of quantiles.

  • 5
    $\begingroup$ If you regard the maximum as the 4th quartile then I'd suggest counting must start with regarding the minimum as the 0th quartile. $\endgroup$
    – Nick Cox
    Jun 13 '15 at 11:52
  • 1
    $\begingroup$ Can percentiles also be scaled to be between 0 and 1? Ex: does it make sense to say percentile(array, 0.5) (the median)? $\endgroup$ Jun 23 '15 at 0:50
  • 2
    $\begingroup$ The "percent" part of "percentile" comes from "cent" for 100. If you scale between 0 and 1 you have proportion. Of course, they are equivalent. $\endgroup$
    – Peter Flom
    Jun 23 '15 at 11:12
  • $\begingroup$ Can you elaborate on "quantiles can go from anything to anything"? I see that in QQ plots, quantiles are not in the [0, 1] range like @stochazesthai's answer says. $\endgroup$
    – arun
    Apr 21 '16 at 2:32
  • 1
    $\begingroup$ @JosephGarvin the point Peter Flom is trying to make here is that quantiles are technically infinitely divisible whereas quartiles are not. E.g. you can have a 11.5625th quantile but only a 1st or 2nd quartile. $\endgroup$
    – jorijnsmit
    May 5 '20 at 21:12

In order to define these terms rigorously, it is helpful to first define the quantile function which is also known as the inverse cumulative distribution function. Recall that for a random variable $X$, the cumulative distribution function $F_X$ is defined by the equation $$ F_X(x) := \Pr(X \le x). $$ The quantile function is defined by the equation $$ Q(p)\,=\,\inf\left\{ x\in \mathbb{R} : p \le F(x) \right\}. $$

Now that we have got these definitions out of the way, we can define the terms:

  • percentile: a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall.

    Example: the 20th percentile of $X$ is the value $Q_X(0.20)$

  • quantile: values taken from regular intervals of the quantile function of a random variable. For instance, for some integer $k \geq 2$, the $k$-quartiles are defined as the values i.e. $Q_X(j/k)$ for $j = 1, 2, \ldots, k - 1$.

    Example: the 5-quantiles of $X$ are the values $Q_X(0.2), Q_X(0.4), Q_X(0.6), Q_X(0.8)$

  • quartile: a special case of quantile, in particular the 4-quantiles. The quartiles of $X$ are the values $Q_X(0.25), Q_X(0.5), Q_X(0.75)$

It may be helpful for you to work out an example of what these definitions mean when say $X \sim U[0,100]$, i.e. $X$ is uniformly distributed from 0 to 100.

References from Wikipedia:

  • 5
    $\begingroup$ Useful, but a very slight awkwardness in the middle. There is no implication in the definition that any discrete set of quantiles you focus on must be selected as regularly spaced in probability. For example, looking at something like 1, 5, 10, 25(25)75, 90, 95, 99 % points is a common part of variable summary. $\endgroup$
    – Nick Cox
    Jun 14 '15 at 13:33
  • $\begingroup$ @NickCox My definition for quantile was to use the definition from Wikipedia en.wikipedia.org/wiki/Quantile "Quantiles are values taken at regular intervals from the inverse of the cumulative distribution function (CDF) of a random variable." $\endgroup$ Jun 15 '15 at 14:14
  • 1
    $\begingroup$ Thanks for the reference, but I contend that using regular intervals is not part of any definition. Quantiles would not cease to be quantiles if you chose (say) 50, 75, 90, 95, 99% points. $\endgroup$
    – Nick Cox
    Jun 15 '15 at 14:49
  • $\begingroup$ @NickCox Your definition makes sense too. I'm not sure why Wikipedia required "regular intervals" in their definition. $\endgroup$ Jun 15 '15 at 18:22
  • 4
    $\begingroup$ I use Wikipedia every day fondly and distrust it mightily on anything like this. $\endgroup$
    – Nick Cox
    Jun 15 '15 at 18:26

From wiki page: https://en.wikipedia.org/wiki/Quantile

Some q-quantiles have special names:

The only 2-quantile is called the median
The 3-quantiles are called tertiles or terciles → T
The 4-quantiles are called quartiles → Q
The 5-quantiles are called quintiles → QU
The 6-quantiles are called sextiles → S
The 8-quantiles are called octiles  → O   (as added by @NickCox - now on wiki page also)
The 10-quantiles are called deciles → D
The 12-quantiles are called duodeciles → Dd
The 20-quantiles are called vigintiles → V
The 100-quantiles are called percentiles → P
The 1000-quantiles are called permilles → Pr

The difference between quantile, quartile and percentile becomes obvious.

  • 4
    $\begingroup$ I've seen also reference to octiles (8). This list is the best argument for the single term quantiles that can be imagined. $\endgroup$
    – Nick Cox
    Jun 13 '15 at 16:50
  • $\begingroup$ I have added it to my answer. You may also add it to wikipedia page. $\endgroup$
    – rnso
    Jun 13 '15 at 17:06
  • 3
    $\begingroup$ Thanks for the edit. I don't think these symbols are anything like standard or even well-chosen; the collective result is just alphabet soup even though it is unlikely that many would be used together. In particular, using $P$ or $Pr$ for anything but a probability is a terrible idea. Who wants to have to remember which way round $Q$ and $Qu$ are? $\endgroup$
    – Nick Cox
    Jun 13 '15 at 17:17
  • 1
    $\begingroup$ I don't participate in writing Wikipedia. Anyone so minded is welcome to add "octile" there. $\endgroup$
    – Nick Cox
    Jun 13 '15 at 17:18

Percentile : The percent of population which lies below that value

Quantile : The cut points dividing the range of probability distribution into continuous intervals with equal probability
There are q-1 of q quantiles one of each k satisfying 0 < k < q

Quartile : Quartile is a special case of quantile, quartiles cut the data set into four equal parts i.e. q=4 for quantiles so we have First quartile Q1, second quartile Q2(Median) and third quartile Q3

  • 2
    $\begingroup$ Your definitions conflict with each other and with standard ones, such as en.wikipedia.org/wiki/Percentile, which make the percentile a particular value of the population rather than a "percent of population." $\endgroup$
    – whuber
    Oct 21 '18 at 20:28
  • $\begingroup$ Percentile is basically the percentage of population lies below that value for example 200 marks in CAT exam is 90 percentile that means 90 percent candidates have marks less than 200 $\endgroup$ Oct 22 '18 at 13:13
  • 1
    $\begingroup$ These definitions really don't add anything helpful to other answers. The definitions of percentile is backward, as @whuber flagged. As flagged elsewhere in the thread the definition of quantile doesn't depend on using equal intervals at all. $\endgroup$
    – Nick Cox
    Aug 22 '20 at 15:29

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