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What is the difference between the three terms below?

  • percentile
  • quantile
  • quartile
askedJun 13, 2015 at 11:25
luciano's user avatar
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    $\begingroup$A deeper question is whether quantiles etc. are intervals or points.$\endgroup$CommentedJun 13, 2015 at 14:20
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    $\begingroup$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.$\endgroup$CommentedJun 13, 2015 at 17:22
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    $\begingroup$My answer atstats.stackexchange.com/questions/235330/… has a fuller list of *ile terms, including dates of first use. Naturally additions and earlier sightings (citings!) are welcome.$\endgroup$CommentedJan 16, 2019 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).$\endgroup$CommentedNov 15, 2020 at 6:19

4 Answers4

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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

answeredJun 13, 2015 at 12:25
stochazesthai's user avatar
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    $\begingroup$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).$\endgroup$CommentedApr 18, 2019 at 22:01
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    $\begingroup$Comparing this answer to one byI Like to Code below, quantile in this answer refers to aquantile function, while another usage ofquantile relates to the division of [0, 1] range of probabilities into equal chunks. n-quantile means division into n chunks$\endgroup$CommentedJan 22, 2022 at 19:33
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  • 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.

answeredJun 13, 2015 at 11:28
Peter Flom's user avatar
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    $\begingroup$If you regard the maximum as the 4th quartile then I'd suggest counting must start with regarding the minimum as the 0th quartile.$\endgroup$CommentedJun 13, 2015 at 11:52
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    $\begingroup$Can percentiles also be scaled to be between 0 and 1? Ex: does it make sense to saypercentile(array, 0.5) (the median)?$\endgroup$CommentedJun 23, 2015 at 0:50
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    $\begingroup$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.$\endgroup$CommentedJun 23, 2015 at 11:12
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    $\begingroup$You can make 1000-tiles or 10,000 tiles or whatever you like.$\endgroup$CommentedApr 22, 2016 at 11:09
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    $\begingroup$@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.$\endgroup$CommentedMay 5, 2020 at 21:12
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In order to define these terms rigorously,it is helpful to first define thequantile functionwhich is also known as theinverse cumulative distribution function.Recall that for a random variable $X$,thecumulative 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 indicatingthe value below which a given percentage of observationsin a group of observations fall.

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

  • quantile: values taken from regular intervalsof the quantile function of a random variable.For instance, for some integer $k \geq 2$,the $k$-quartiles are defined as the valuesi.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 meanwhen say $X \sim U[0,100]$,i.e. $X$ is uniformly distributed from 0 to 100.

References from Wikipedia:

answeredJun 13, 2015 at 20:32
I Like to Code's user avatar
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    $\begingroup$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.$\endgroup$CommentedJun 14, 2015 at 13:33
  • $\begingroup$@NickCox My definition for quantile was to use the definition from Wikipediaen.wikipedia.org/wiki/Quantile "Quantiles are values taken at regular intervals from the inverse of the cumulative distribution function (CDF) of a random variable."$\endgroup$CommentedJun 15, 2015 at 14:14
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    $\begingroup$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.$\endgroup$CommentedJun 15, 2015 at 14:49
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    $\begingroup$I use Wikipedia every day fondly and distrust it mightily on anything like this.$\endgroup$CommentedJun 15, 2015 at 18:26
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    $\begingroup$If a specify an arbitrary value p in the interval [0 1] in the definition of Q(p) above and want to find the corresponding x, would x be called the p-quantile?$\endgroup$CommentedAug 31, 2021 at 18:13
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From wiki page:https://en.wikipedia.org/wiki/Quantile

Some q-quantiles have special names:

The only 2-quantile is called the medianThe 3-quantiles are called tertiles or terciles → TThe 4-quantiles are called quartiles → QThe 5-quantiles are called quintiles → QUThe 6-quantiles are called sextiles → SThe 8-quantiles are called octiles  → O   (as added by @NickCox - now on wiki page also)The 10-quantiles are called deciles → DThe 12-quantiles are called duodeciles → DdThe 20-quantiles are called vigintiles → VThe 100-quantiles are called percentiles → PThe 1000-quantiles are called permilles → Pr

The difference betweenquantile,quartile andpercentile becomes obvious.

answeredJun 13, 2015 at 12:24
rnso's user avatar
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    $\begingroup$I've seen also reference to octiles (8). This list is the best argument for the single termquantiles that can be imagined.$\endgroup$CommentedJun 13, 2015 at 16:50
  • $\begingroup$I have added it to my answer. You may also add it to wikipedia page.$\endgroup$CommentedJun 13, 2015 at 17:06
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    $\begingroup$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?$\endgroup$CommentedJun 13, 2015 at 17:17
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    $\begingroup$I don't participate in writing Wikipedia. Anyone so minded is welcome to add "octile" there.$\endgroup$CommentedJun 13, 2015 at 17:18
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