A quantile of a distribution is a value that divides its probability mass in a specified proportion. If
a random variable 
has distribution function 
, a standard quantile
function is the generalized inverse defined, for 
, by
 |
(1)
|
When 
is continuous
and strictly increasing, 
is the unique value 
satisfying 
. For a discontinuous distribution,
such an equality need not have a solution, which is why the generalized-inverse definition
is used (Hyndman and Fan 1996).
The 
th 
-tile is the quantile 
for 
, ..., 
(Kenney and Keeping 1962). If 
, the interior quantiles are called quartiles,
and if 
, they are called percentiles.
The standard names for several common divisions are summarized below. Dividing a distribution into 
equal parts requires 
interior cut points. In particular, the two cut points that
divide a distribution into thirds are tertiles,
while the three resulting groups are terciles.
| number of parts | cut-point name | number of cut points |
| 3 | tertile | 2 |
| 4 | quartile | 3 |
| 5 | quintile | 4 |
| 10 | decile | 9 |
| 100 | percentile | 99 |
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/Inline26.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 
.
