- numpy.quantile(
*a*,*q*,*axis=None*,*out=None*,*overwrite_input=False*,*method='linear'*,*keepdims=False*,***,*weights=None*,*interpolation=None*)[source]# Compute the q-th quantile of the data along the specified axis.

New in version 1.15.0.

- Parameters:
**a**array_like of real numbersInput array or object that can be converted to an array.

**q**array_like of floatProbability or sequence of probabilities of the quantiles to compute.Values must be between 0 and 1 inclusive.

**axis**{int, tuple of int, None}, optionalAxis or axes along which the quantiles are computed. The default isto compute the quantile(s) along a flattened version of the array.

**out**ndarray, optionalAlternative output array in which to place the result. It must havethe same shape and buffer length as the expected output, but thetype (of the output) will be cast if necessary.

**overwrite_input**bool, optionalIf True, then allow the input array

*a*to be modified byintermediate calculations, to save memory. In this case, thecontents of the input*a*after this function completes isundefined.**method**str, optionalThis parameter specifies the method to use for estimating thequantile. There are many different methods, some unique to NumPy.The recommended options, numbered as they appear in [1], are:

‘inverted_cdf’

‘averaged_inverted_cdf’

‘closest_observation’

‘interpolated_inverted_cdf’

‘hazen’

‘weibull’

‘linear’ (default)

‘median_unbiased’

‘normal_unbiased’

The first three methods are discontinuous. For backward compatibilitywith previous versions of NumPy, the following discontinuous variationsof the default ‘linear’ (7.) option are available:

See Alsoquantile function - RDocumentationQuantile() function in R - A brief guide | DigitalOceanCentron Tutorials – Expertenwissen für Cloud-Technologien und IT-Infrastrukturquantile: Sample Quantiles‘lower’

‘higher’,

‘midpoint’

‘nearest’

See Notes for details.

Changed in version 1.22.0: This argument was previously called “interpolation” and onlyoffered the “linear” default and last four options.

**keepdims**bool, optionalIf this is set to True, the axes which are reduced are left inthe result as dimensions with size one. With this option, theresult will broadcast correctly against the original array

*a*.**weights**array_like, optionalAn array of weights associated with the values in

*a*. Each value in*a*contributes to the quantile according to its associated weight.The weights array can either be 1-D (in which case its length must bethe size of*a*along the given axis) or of the same shape as*a*.If*weights=None*, then all data in*a*are assumed to have aweight equal to one.Only*method=”inverted_cdf”*supports weights.See the notes for more details.New in version 2.0.0.

**interpolation**str, optionalDeprecated name for the method keyword argument.

Deprecated since version 1.22.0.

- Returns:
**quantile**scalar or ndarrayIf

*q*is a single probability and*axis=None*, then the resultis a scalar. If multiple probability levels are given, first axisof the result corresponds to the quantiles. The other axes arethe axes that remain after the reduction of*a*. If the inputcontains integers or floats smaller than`float64`

, the outputdata-type is`float64`

. Otherwise, the output data-type is thesame as that of the input. If*out*is specified, that array isreturned instead.

See also

- mean
- percentile
equivalent to quantile, but with q in the range [0, 100].

- median
equivalent to

`quantile(..., 0.5)`

- nanquantile

Notes

Given a sample

*a*from an underlying distribution, quantile provides anonparametric estimate of the inverse cumulative distribution function.By default, this is done by interpolating between adjacent elements in

`y`

, a sorted copy of*a*:(1-g)*y[j] + g*y[j+1]

where the index

`j`

and coefficient`g`

are the integral andfractional components of`q * (n-1)`

, and`n`

is the number ofelements in the sample.This is a special case of Equation 1 of H&F [1]. More generally,

`j = (q*n + m - 1) // 1`

, and`g = (q*n + m - 1) % 1`

,

where

`m`

may be defined according to several different conventions.The preferred convention may be selected using the`method`

parameter:`method`

number in H&F

`m`

`interpolated_inverted_cdf`

4

`hazen`

5

`1/2`

`weibull`

6

`q`

`linear`

(default)7

`1 - q`

`median_unbiased`

8

`q/3 + 1/3`

`normal_unbiased`

9

`q/4 + 3/8`

Note that indices

`j`

and`j + 1`

are clipped to the range`0`

to`n - 1`

when the results of the formula would be outside the allowedrange of non-negative indices. The`- 1`

in the formulas for`j`

and`g`

accounts for Python’s 0-based indexing.The table above includes only the estimators from H&F that are continuousfunctions of probability

*q*(estimators 4-9). NumPy also provides thethree discontinuous estimators from H&F (estimators 1-3), where`j`

isdefined as above,`m`

is defined as follows, and`g`

is a functionof the real-valued`index = q*n + m - 1`

and`j`

.`inverted_cdf`

:`m = 0`

and`g = int(index - j > 0)`

`averaged_inverted_cdf`

:`m = 0`

and`g = (1 + int(index - j > 0)) / 2`

`closest_observation`

:`m = -1/2`

and`g = 1 - int((index == j) & (j%2 == 1))`

For backward compatibility with previous versions of NumPy, quantileprovides four additional discontinuous estimators. Like

`method='linear'`

, all have`m = 1 - q`

so that`j = q*(n-1) // 1`

,but`g`

is defined as follows.`lower`

:`g = 0`

`midpoint`

:`g = 0.5`

`higher`

:`g = 1`

`nearest`

:`g = (q*(n-1) % 1) > 0.5`

**Weighted quantiles:**More formally, the quantile at probability level \(q\) of a cumulativedistribution function \(F(y)=P(Y \leq y)\) with probability measure\(P\) is defined as any number \(x\) that fulfills the*coverage conditions*\[P(Y < x) \leq q \quad\text{and}\quad P(Y \leq x) \geq q\]

with random variable \(Y\sim P\).Sample quantiles, the result of quantile, provide nonparametricestimation of the underlying population counterparts, represented by theunknown \(F\), given a data vector

*a*of length`n`

.Some of the estimators above arise when one considers \(F\) as theempirical distribution function of the data, i.e.\(F(y) = \frac{1}{n} \sum_i 1_{a_i \leq y}\).Then, different methods correspond to different choices of \(x\) thatfulfill the above coverage conditions. Methods that follow this approachare

`inverted_cdf`

and`averaged_inverted_cdf`

.For weighted quantiles, the coverage conditions still hold. Theempirical cumulative distribution is simply replaced by its weightedversion, i.e. \(P(Y \leq t) = \frac{1}{\sum_i w_i} \sum_i w_i 1_{x_i \leq t}\).Only

`method="inverted_cdf"`

supports weights.References

[1](1,2)

R. J. Hyndman and Y. Fan,“Sample quantiles in statistical packages,”The American Statistician, 50(4), pp. 361-365, 1996

Examples

>>> import numpy as np>>> a = np.array([[10, 7, 4], [3, 2, 1]])>>> aarray([[10, 7, 4], [ 3, 2, 1]])>>> np.quantile(a, 0.5)3.5>>> np.quantile(a, 0.5, axis=0)array([6.5, 4.5, 2.5])>>> np.quantile(a, 0.5, axis=1)array([7., 2.])>>> np.quantile(a, 0.5, axis=1, keepdims=True)array([[7.], [2.]])>>> m = np.quantile(a, 0.5, axis=0)>>> out = np.zeros_like(m)>>> np.quantile(a, 0.5, axis=0, out=out)array([6.5, 4.5, 2.5])>>> marray([6.5, 4.5, 2.5])>>> b = a.copy()>>> np.quantile(b, 0.5, axis=1, overwrite_input=True)array([7., 2.])>>> assert not np.all(a == b)

See also numpy.percentile for a visualization of most methods.

# numpy.quantile — NumPy v2.1 Manual (2024)

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