Percentile vs quantile vs quartile What is the difference between the three terms below?


*

*percentile

*quantile

*quartile

 A: *

*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. 
A: 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:


*

*https://en.wikipedia.org/wiki/Quantile

*https://en.wikipedia.org/wiki/Quantile_function

*https://en.wikipedia.org/wiki/Quartile

*https://en.wikipedia.org/wiki/Percentile
A: 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.
A: 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
