Quantile

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

(2)

where  is the th order statistic,  is the floor function,  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 .

REFERENCES:

Barnett, V. "Probability Plotting Methods and Order Statistics." Appl. Stat. 24, 95-108, 1975.

Cunnane, C. "Unbiased Plotting Positions--A Review." J. Hydrology 37, 205-222, 1978.

Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, 2000.

Harter, H. L. "Another Look at Plotting Positions." Comm. Stat., Th. and Methods 13, 1613-1633, 1984.

Hyndman, R. J. and Fan, Y. "Sample Quantiles in Statistical Packages." Amer. Stat. 50, 361-365, 1996.

Kenney, J. F. and Keeping, E. S. "Quantiles." §3.5 in Mathematics of Statistics, Pt. 1, 3rd ed. Princeton, NJ: Van Nostrand, pp. 37-38, 1962.