Detailed algorithms

Eleven quantile algorithms are made available: 9 are discussed in Hyndman and Fan’s, 1 in Cunnane’s and 1 in Filliben’s articles (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

1. Makkonen L. and Pajari M. (2014): Defining sample quantiles by the true rank probability, Journal of Probability and Statistics, vol. 2014, Article ID 326579, doi:10.1155/2014/326579
2. Hyndman R.J. and Fan Y. (1996): Sample quantiles in statistical packages, The American Statistician, 50(4):361-365, doi:10.2307/2684934
3. Cunnane C. (1978): Unbiased plotting positions: a review, Journal of Hydrology, 37(3-4):205-222, doi:10.1016/0022-1694(78)90017-3.
4. Barnett V. (1975): Probability plotting methods and order statistics, Journal of the Royal Statistical Society. Series C (Applied Statistics), 24(1):95-108, doi:10.2307/2346708 .
5. Filliben J.J. (1975): The probability plot correlation coefficient test for normality, Technometrics, 17(1):111-117, doi:10.2307/1268008.