The word quantile has no fewer than two distinct meanings in probability. Specific elements 
in the range of a variate 
are called quantiles, and denoted 
(Evans et al. 2000, p. 5).
This particular meaning has close ties to the so-called quantile
function, a function which assigns to each probability 
attained by a certain probability
density function 
a value 
defined by
 |
(1)
|
The 
th 
-tile 
is that value of 
,
say 
, which corresponds to a cumulative
frequency of 
(Kenney and Keeping 1962). If 
, the quantity is called a quartile,
and if 
, it is called a percentile.
A parametrized version of quantile is implemented as Quantile[list, q, 
a, b
, 
c,
d
], which returns
![q_(a,b;c,d)(X_1,...,X_N)=Y_(|_x_|)+(Y_([x])-Y_(|_x_|))(c+dfrac(x)),](/images/equations/Quantile/NumberedEquation2.svg) |
(2)
|
where 
is the 
th order statistic, 
is the floor
function, ![[x]](/images/equations/Quantile/Inline24.svg)
is the ceiling function, 
is the fractional part,
and
 |
(3)
|
There are a number of slightly different definitions of the quantile that are in common use, as summarized in the following table.
| # |  |  |  |  | plotting position | description |
| Q1 | 0 | 0 | 1 | 0 |  | inverted empirical CDF |
| Q2 | -- | -- | -- | -- |  | inverted empirical CDF with averaging |
| Q3 |  | 0 | 0 | 0 |  | observation
numberer closest to  |
| Q4 | 0 | 0 | 0 | 1 |  | California Department of Public Works method |
| Q5 |  | 0 | 0 | 1 |  | Hazen's model (popular in hydrology) |
| Q6 | 0 | 1 | 0 | 1 |  | Weibull quantile |
| Q7 | 1 |  | 0 | 1 |  | interpolation points divide sample range into  intervals |
| Q8 |  |  | 0 | 1 |  | unbiased median |
| Q9 |  |  | 0 | 1 |  | approximate unbiased estimate for a normal distribution |
The Wolfram Language's parametrization can handle all of these but Q2. In Q1, the empirical distribution
function is the estimated cumulative proportion of the data set that does not
exceed any specified value. Q2 is essentially the same as Q1 except that averages
are taken at points of discontinuity. In Q3, the 
th quantile is the observation numbered closest to 
, where 
is the sample size. In Q4,
the interpolation points divide the sample range into 
intervals. In Q6, the vertices divide the sample into 
regions, each with probability 
on average. It was proposed by
Weibull in 1939, and plots 
at the mean position. Q7 divides the range into 
intervals, of which exactly 
lie to the left of 
. Q8 plots 
at the median position. Q9 is used in quantile-quantile
plots. If 
is the normal distribution and 
is the plotting position
of 
, then 
is an approximately unbiased estimate of 
.
