Quantile
Details
- Quantile is also known as value at risk (VaR) or fractile.
- When VectorQ data is sorted as , the quantile estimate is given by .
- For MatrixQ data, the quantile is computed for each column vector with Quantile [{{x1,y1,…},{x2,y2,…},…},p] equivalent to {Quantile[{x1,x2,…},p],Quantile[{y1,y2,…},p]}. »
- For ArrayQ data, quantile is equivalent to ArrayReduce [Quantile ,data,1]. »
- Quantile [{x_1,...,x_n},p,{{a,b},{c,d}}] is given by with r=a+(n+b)p, ⌊r⌋= Floor [r], ⌈r⌉=Ceiling [r] and =FractionalPart [r]. The indices are taken to be 1 or n if they are out of range. »
- Common choices of parameters {{a,b},{c,d}} include:
-
{{0,0},{1,0}} inverse empirical CDF (default){{0,0},{0,1}} linear interpolation (California method){{1/2,0},{0,0}} element numbered closest to p n{{1/2,0},{0,1}} linear interpolation (hydrologist method){{0,1},{0,1}} mean‐based estimate (Weibull method){{1,-1},{0,1}} mode‐based estimate{{1/3,1/3},{0,1}} median‐based estimate{{3/8,1/4},{0,1}} normal distribution estimate
- The default choice of parameters is {{0,0},{1,0}}.
- About 10 different choices of parameters are in use in statistical work.
- Quantile [list,p] always gives a result equal to an element of list.
- The same is true whenever d is 0.
- When d is 1, Quantile is piecewise linear as a function of p.
- Median [data] is equivalent to Quantile [data,1/2,{{1/2,0},{0,1}}].
- The data can have the following additional forms and interpretations:
-
Association the values (the keys are ignored) »SparseArray as an array, equivalent to Normal [data] »QuantityArray quantities as an array »WeightedData based on the underlying EmpiricalDistribution »EventData based on the underlying SurvivalDistribution »
- Quantile [dist,p] is equivalent to InverseCDF [dist,p].
- Quantile [dist,p] is the minimum of the set of number(s) q_(p) such that Probability [x≤q_(p),xdist]≥p and Probability [x≥q_(p),xdist]≥p. »
- For a random process proc, the quantile function can be computed for slice distribution at time t, SliceDistribution [proc,t], as Quantile [SliceDistribution [proc,t], p]. »
- The value p can be symbolic or any number between 0 and 1. »
Examples
open all close allBasic Examples (7)
Find the halfway value of a list:
Find the 20% and 80% quantiles of a list:
Find the top percentile of a list:
Quantile of a list of dates:
The q^(th) quantile for a normal distribution:
Quantile function for a continuous univariate distribution:
Quantile function for a discrete univariate distribution:
Scope (33)
Basic Uses (7)
Quantile works with any real numeric quantities:
Obtain results at any precision:
Compute results using other parametrizations:
Find quantiles for WeightedData :
Find quantiles for EventData :
Find a quantile for TimeSeries :
The quantile depends only on the values:
Find a quantile for data involving quantities:
Array Data (6)
Find quantiles of elements in each column:
Find multiple quantiles of elements in each column:
The quantile for a tensor gives columnwise standard deviations at the first level:
Compute results for a large vector or matrix:
When the input is an Association , Quantile works on its values:
Compute results for a SparseArray :
Find a quantile of a QuantityArray :
Image and Audio Data (2)
Channelwise 30% percentile value of an RGB image:
30% percentile intensity value of a grayscale image:
30% percentile amplitude of all channels:
Date and Time (5)
Compute a quantile of dates:
Compute a weighted quantile of dates:
Compute a quantile of dates given in different calendars:
The quantile is given in one of the input calendars:
Compute a quantile of times:
Compute a quantile of times with different time zone specifications:
Parametric Distributions (5)
Obtain exact numeric results:
Obtain a machine-precision result:
Obtain a result at any precision for a continuous distribution:
Obtain a symbolic expression for the quantile:
Quantile threads elementwise over lists:
Nonparametric Distributions (2)
Quantile for nonparametric distributions:
Compare with the value for the underlying parametric distribution:
Plot the quantile for a histogram distribution:
Derived Distributions (4)
Quantile for a truncated distribution:
Quadratic transformation of an exponential distribution:
Censored distribution:
Quantile for distributions with quantities:
Random Processes (2)
Quantile function for a random process:
Find a quantile of TemporalData at some time t=0.5:
Find the corresponding quantile function together with all the simulations:
Applications (7)
A set of equally spaced quantiles divides the values into equal-sized groups:
Calculate a set of quantiles:
Plot the PDF divided according to the values of quantiles into five regions:
Use quantile as a mesh function:
Plot the q^(th) quantile for a list:
The linearly interpolated quantile:
Compute an expectation using quantile :
Use this method in Expectation :
Generate random numbers for a nonuniform distribution by transforming the uniform distribution by the quantile function of the nonuniform distribution:
Compare the histogram of the sample with the probability density function of the desired distribution:
Compute a moving quantile for some data:
Use the window of length .1:
Compute selected quantiles for slices of a collection of paths of a random process:
Choose a few slice times:
Plot the quantiles over these paths:
Compute quantiles for the heights of children in a class:
Properties & Relations (9)
Use Quantile to find the quartiles of a distribution:
Calculate quartiles directly:
With default parameters, Quantile always returns an element of the list:
Quartiles gives linearly interpolated Quantile values for a list:
InterquartileRange is the difference of linearly interpolated Quantile values for a list:
QuartileDeviation is half the difference of linearly interpolated Quantile values for a list:
QuartileSkewness uses linearly interpolated Quantile values as a skewness measure:
Quantile is equivalent to InverseCDF for distributions:
QuantilePlot plots the quantiles of a list or distribution:
BoxWhiskerChart shows special quantiles for data:
Possible Issues (4)
For computations with data, the value p can be any number between 0 and 1:
The symbolic closed form may exist for some distributions:
Symbolic closed forms do not exist for some distributions:
Numerical evaluation works:
Substitution of invalid values into symbolic outputs gives results that are not meaningful:
It stays unevaluated if passed as an argument:
Quartiles of data computed via Quantile do not always agree with Quartiles :
Calculate quartiles directly:
Specify linear interpolation parameters in Quantile :
Neat Examples (1)
The distribution of Quantile estimates for 20, 100, and 300 samples:
See Also
Median Quartiles Ordering Variance MedianDeviation InterquartileRange Sort ListInterpolation Nearest InverseCDF InverseSurvivalFunction OrderDistribution
Function Repository: StatisticsSummary
Tech Notes
History
Introduced in 2003 (5.0) | Updated in 2007 (6.0) ▪ 2023 (13.3) ▪ 2024 (14.1)
Text
Wolfram Research (2003), Quantile, Wolfram Language function, https://reference.wolfram.com/language/ref/Quantile.html (updated 2024).
CMS
Wolfram Language. 2003. "Quantile." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/Quantile.html.
APA
Wolfram Language. (2003). Quantile. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Quantile.html
BibTeX
@misc{reference.wolfram_2025_quantile, author="Wolfram Research", title="{Quantile}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/Quantile.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_quantile, organization={Wolfram Research}, title={Quantile}, year={2024}, url={https://reference.wolfram.com/language/ref/Quantile.html}, note=[Accessed: 04-January-2026]}