Divide a given sample, possibly weighted, into a certain number of slices of equal size, with units ranked according to a variable of interest.

Contents:

Algorithm

Nine quantile algorithms are made available, as discussed in Hyndman and Fan's, plus two additional coming also from the literature, namely Filliben’s and Cunnane's publication (see references):

`type`	description
1	inverted empirical CDF
2	inverted empirical CDF with averaging at discontinuities
3	observation numberer closest to qN (piecewise linear function)
4	linear interpolation of the empirical CDF
5	Hazen's model (piecewise linear function)
6	Weibull quantile
7	interpolation points divide sample range into n-1 intervals
8	unbiased median (regardless of the distribution)
9	approximate unbiased estimate for a normal distribution
10	Cunnane's definition (approximately unbiased)
11	Filliben’s estimate

All sample quantiles are defined as weighted averages of consecutive order statistics. Sample quantiles of type i are defined for 1 <= i <= 10 by:

Q[i](p) = (1 - gamma) * x[j] + gamma *  x[j+1]

where x[j], for (j-m)/N<=p<(j-m+1)/N, is the j-th order statistic, N is the sample size, the value of gamma is a function of:

j = floor(N*p + m)
g = N*p + m - j

and m is a constant determined by the sample quantile type.

For types 1, 2 and 3, Q[i](p) is a discontinuous function:

`type`	`p[k]`	`m`	`alphap`	`betap`	`gamma`
1	`k/N`	0	0	1	1 if `g>0`, 0 if `g=0`
2	`k/N`	0	0	1	1/2 if `g>0`, 0 if `g=0`
3	`(k+1/2)/N`	-.5	-.5	-1.5	0 if `g=0` and `j` even, 1 otherwise

For types 4 through 11, Q[i](p) is a continuous function of p, with gamma and m given below. The sample quantiles can be obtained equivalently by linear interpolation between the points (p[k],x[k]) where x[k] is the k-th order statistic:

`type`	`p[k]`	`m`	`alphap`	`betap`	`gamma`
4	`k/N`	0	0	1	`g`
5	`(k-1/2)/N`	.5	.5	.5	`g`
6	`k/(N+1)`	`p`	0	0	`g`
7	`(k-1)/(N-1)`	`1-p`	1	1	`g`
8	`(k-1/3)/(N+1/3)`	`(1+p)3`	1/3	1/3	`g`
9	`(k-3/8)/(N+1/4)`	`(2*p+3)/8`	3/8	3/8	`g`
10	`(k-.4)/(N+.2)`	`.2*p+.4`	.4	.4	`g`
11	`(k-.3175)/(N+.365)`	`.365*p+.3175`	.3175	.3175	`g`

In the above tables, the (alphap,betap) pair is defined such that:

p[k] = (k - alphap)/(N + 1 - alphap - betap)

References

Makkonen, L. and Pajari, M. (2014): "Defining sample quantiles by the true rank probability".
Hyndman, R.J. and Fan, Y. (1996): Sample quantiles in statistical packages, The American Statistician, 50(4):361-365, doi:10.2307/2684934
Cunnane, C. (1978): Unbiased plotting positions: a review, Journal of Hydrology, 37(3-4):205-222, doi:10.1016/0022-1694(78)90017-3.
Filliben, J.J. (1975): The probability plot correlation coefficient test for normality, Technometrics, 17(1):111-117, doi:10.2307/1268008.
Barnett, V. (1975): "Probability plotting methods and order statistics".