Syntax

quantile: Compute empirical quantiles of a variable with sample data corresponding to given probabilities.

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Common parameterisation

Some arguments are common to the implementations in the different languages:

• probs : (option) list of probabilities with values in [0,1]; the smallest observation corresponds to a probability of 0 and the largest to a probability of 1; default: probs is set to the sequence 0 0.25 0.5 0.75 1, so as to match default values seq(0, 1, 0.25) used in R quantile;

• type : (option) an integer between 1 and 11 selecting one of the 9+1+1 quantile algorithms discussed in Hyndman and Fan’s, Cunane’s and Filliben’s articles (see references) and detailed below to be used;

  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

  default: type=7 (likewise R quantile);

• method : (option) choice of the implementation of the quantile estimation method; this can be either:
  • INHERIT for an estimation based on the use of an already existing implementation in the given language,
  • DIRECT for a canonical implementation based on the direct transcription of the various quantile estimation algorithms (see below) into the given language;

  default: method=DIRECT;

• na_rm : (option) logical; if true, any NA and NaN’s are removed from x before the quantiles are computed.

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SAS macro

%quantile(var, probs=, type=7, method=DIRECT, names=, _quantiles_=, 
	  idsn=, odsn=, ilib=WORK, olib=WORK, na_rm = YES);

Arguments

• var : data whose sample quantiles are estimated; this can be either:
  • the name of the variable in a dataset storing the data; in that case, the parameter idsn (see below) should be set;
  • a list of (blank separated) numeric values;
• idsn : (option) when input data is passed as a variable name, idsn represents the dataset to look for the variable var (see above);
• ilib : (option) name of the input library; by default: empty, i.e. WORK is used if idsn is set;
• olib : (option) name of the output library (see names below); by default: empty, i.e. WORK is also used when odsn is set.

Returns Return estimates of underlying distribution quantiles based on one or two order statistics from the supplied elements in var at probabilities in probs, following quantile estimation algorithm defined by type. The output sample quantile are stored either in a list or as a table, through:

• _quantiles_ : (option) name of the output numeric list where quantiles are stored in increasing probs order; incompatible with parameters odsn and names below;
• odsn, names : (option) respective names of the output dataset and variable where quantiles are stored; if both odsn and names are set, the quantiles are saved in the names variable ot the odsn dataset; if just odsn is set, then they are stored in a variable named QUANT; if instead only names is set, then the dataset will also be named after names.

Notes

• probs : (see above) in the case method=UNIVAR (see below), these values are multiplied by 100 in order to be used by PROC UNIVARIATE;
• method : (see above) in the case method=INHERIT, the macro uses the PROC UNIVARIATE procedure already implemented in SAS; this is incompatible with type other than (1,2,3,4,6) since PROC UNIVARIATE does actually not support these quantile definitions (see table above); in the case type=5, 7, 8, or 9, method is then set to DIRECT.
• type : (see above) note the (non bijective) correspondance between the different algorithms and the currently available methods in PROC UNIVARIATE (through the use of PCTLDEF parameter):

type	1	2	3	4	5	6	7	8	9	10	11
PCTLDEF	3	5	2	1	n.a.	4	n.a.	n.a.	n.a.	n.a.	n.a.

• na_rm : (see above) true is yes, false is no.

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Python method

>>> q = quantile(x, probs, na_rm = False, type = 7, 
	  method='DIRECT', limit=(0,1))

Arguments

• x : input 1D (vector) data (numpy.array, pandas.DataFrame, or pandas.Series); 2D arrays are also accepted;
• limit : (option) tuple/list of (lower, upper) values; values of a outside this open interval are ignored.

Returns

• q : 1D vector of quantiles returned as a numpy.array.

Notes

• probs : (see above) the following codes: 2 or M2, 3 or T3, 4 or Qu4, 5 or Q5, 6 or S6, 10 or D10, 12 or Dd12, 20 or V20, and 100 or P100` can be used to compute common specialised quantiles (median, terciles, quartiles, quintiles, sextiles, deciles, duo-deciles, ventiles and percentiles resp.);
• method : (see above) in the case method=INHERIT, the scipy::mquantiles function is used to estimate quantiles; this case is incompatible with type<4 (see below);
• type : (see above) methods 4 to 11 are available in original scipy::mquantiles function;
• na_rm : (see above) true is True, false is False.

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R method

> q <- quantile(x, data = NULL, probs=seq(0, 1, 0.25), na.rm=FALSE, 
	  type=7, method="DIRECT", names= FALSE)

Arguments

• x : a numeric vector or a value (character or integer) providing with the sample data; when data is not null, x provides with the name (char) or the position (int) of the variable of interest in the table;
• data : (option) input table, defined as a dataframe, whose column defined by x is used as sample data for the estimation; if passed, then x should be defined as a character or an integer; default: data=NULL and input sample data should be passed as numeric vector in x;
• probs : (option) numeric vector giving the probabilities with values in [0,1]; default: probs=seq(0, 1, 0.25) like in original stats::quantile function;
• na_rm, names : (option) logical flags; if na.rm=TRUE, any NA and NaN’s are removed from x before the quantiles are computed; if names=TRUE, the result has a names attribute; these two flags follow exactly the original implementation of stats::quantile; default: na.rm= FALSE and names= FALSE.

Notes

• method : (see above) in the case method=INHERIT, the stats::quantile function is used to estimate quantiles; this case is incompatible with type>9 (see below);
• type : (see above) methods 1 to 9 are available in original stats::quantile function.

Returns

• q : 1D vector of quantiles returned as a numeric vector.

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See also

• UNIVARIATE.
• quantile (R).
• mquantiles (scipy).
• gsl_stats_quantile* (C).