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Quantile

Quantile[data,p]

gives the estimate of the p quantile of data.

Quantile[data,{p₁,p₂,…}]

gives a list of quantiles p₁,p₂,….

Quantile[data,p,{{a,b},{c,d}}]

uses the quantile definition specified by parameters a, b, c, d.

Quantile[dist,p]

gives a quantile of the distribution dist.

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 »
	TimeSeries, TemporalData, …	vector or array of values (the time stamps ignored) »
	Image,Image3D	RGB channel's values or grayscale intensity value »
	Audio	amplitude values of all channels »
	DateObject, TimeObject	list of dates or list of times »

Quantile[dist,p] is equivalent to InverseCDF[dist,p].
Quantile[dist,p] is the minimum of the set of number(s) such that Probability[x≤,xdist]≥p and Probability[x≥,xdist]≥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

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Basic 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 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 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:

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Quantile

Details

Examples

Basic Examples (7)