# quantile

Quantiles of data set

## Syntax

## Description

returns quantiles of elements in input data `Q`

= quantile(`A`

,`p`

)`A`

for the cumulative
probability or probabilities `p`

in the interval [0,1].

If

`A`

is a vector, then`Q`

is a scalar or a vector with the same length as`p`

.`Q(i)`

contains the`p(i)`

quantile.If

`A`

is a matrix, then`Q`

is a row vector or a matrix, where the number of rows of`Q`

is equal to`length(p)`

. The`i`

th row of`Q`

contains the`p(i)`

quantiles of each column of`A`

.If

`A`

is a multidimensional array, then`Q`

contains the quantiles computed along the first array dimension of size greater than 1.

returns quantiles for `Q`

= quantile(`A`

,`n`

)`n`

evenly spaced cumulative probabilities
(1/(`n`

+ 1), 2/(`n`

+ 1), ...,
`n`

/(`n`

+ 1)) for integer `n`

> 1.

If

`A`

is a vector, then`Q`

is a scalar or a vector with length`n`

.If

`A`

is a matrix, then`Q`

is a matrix with`n`

rows.If

`A`

is a multidimensional array, then`Q`

contains the quantiles computed along the first array dimension of size greater than 1.

returns
quantiles of all the elements of `Q`

= quantile(___,"all")`A`

for either of the first two
syntaxes.

operates along the dimension `Q`

= quantile(___,`dim`

)`dim`

for either of the first two syntaxes.
For example, if `A`

is a matrix, then `quantile(A,p,2)`

operates on the elements in each row.

operates along the dimensions specified in the vector `Q`

= quantile(___,`vecdim`

)`vecdim`

for either
of the first two syntaxes. For example, if `A`

is a matrix, then
`quantile(A,n,[1 2])`

operates on all the elements of
`A`

because every element of a matrix is contained in the array slice
defined by dimensions 1 and 2.

## Examples

### Quantiles for Given Probabilities

Calculate the quantiles of a data set for specified probabilities.

Generate a data set of size 7.

rng default % for reproducibility A = randn(1,7)

`A = `*1×7*
0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336

Calculate the 0.3 quantile of the elements of `A`

.

Q = quantile(A,0.3)

Q = -0.7832

Calculate the quantiles of the elements of `A`

for the cumulative probabilities 0.025, 0.25, 0.5, 0.75, and 0.975.

Q = quantile(A,[0.025 0.25 0.5 0.75 0.975])

`Q = `*1×5*
-2.2588 -1.0892 0.3188 0.7810 1.8339

### Quantiles for `n`

Evenly Spaced Cumulative Probabilities

Calculate the quantiles of a data set for a given number of probabilities.

Generate a data set of size 7.

rng default % for reproducibility A = randn(1,7)

`A = `*1×7*
0.5377 1.8339 -2.2588 0.8622 0.3188 -1.3077 -0.4336

Calculate four evenly spaced quantiles of the elements of `A`

.

Q = quantile(A,4)

`Q = `*1×4*
-1.4028 -0.2079 0.4720 0.9593

Using `Q = quantile(A,[0.2,0.4,0.6,0.8])`

is another way to return the four evenly spaced quantiles.

### Quantiles of Matrix for Given Probabilities

Calculate the quantiles along the columns and rows of a data matrix for specified probabilities.

Generate a 4-by-6 data matrix.

rng default % for reproducibility A = randn(4,6)

`A = `*4×6*
0.5377 0.3188 3.5784 0.7254 -0.1241 0.6715
1.8339 -1.3077 2.7694 -0.0631 1.4897 -1.2075
-2.2588 -0.4336 -1.3499 0.7147 1.4090 0.7172
0.8622 0.3426 3.0349 -0.2050 1.4172 1.6302

Calculate the 0.3 quantile for each column of `A`

.

Q = quantile(A,0.3,1)

`Q = `*1×6*
-0.3013 -0.6958 1.5336 -0.1056 0.9491 0.1078

`quantile`

returns a row vector `Q`

when calculating one quantile for each column in `A`

. `-0.3013`

is the 0.3 quantile of the first column of `A`

with elements 0.5377, 1.8339, -2.2588, and 0.8622. Because the default value of `dim`

is 1, `Q = quantile(A,0.3)`

returns the same result.

Calculate the 0.3 quantile for each row of `A`

.

Q = quantile(A,0.3,2)

`Q = `*4×1*
0.3844
-0.8642
-1.0750
0.4985

`quantile`

returns a column vector `Q`

when calculating one quantile for each row in `A`

. `0.3844`

is the 0.3 quantile of the first row of `A`

with elements 0.5377, 0.3188, 3.5784, 0.7254, -0.1241, and 0.6715.

### Quantiles of Matrix for `n`

Evenly Spaced Probabilities

Calculate evenly spaced quantiles along the columns and rows of a data matrix.

Generate a 6-by-7 data matrix.

rng default % for reproducibility A = randi(10,6,7)

`A = `*6×7*
9 3 10 8 7 8 7
10 6 5 10 8 1 4
2 10 9 7 8 3 10
10 10 2 1 4 1 1
7 2 5 9 7 1 5
1 10 10 10 2 9 4

Calculate the quantiles for each column of `A`

for three evenly spaced cumulative probabilities.

Q = quantile(A,3,1)

`Q = `*3×7*
2.0000 3.0000 5.0000 7.0000 4.0000 1.0000 4.0000
8.0000 8.0000 7.0000 8.5000 7.0000 2.0000 4.5000
10.0000 10.0000 10.0000 10.0000 8.0000 8.0000 7.0000

Each column of matrix `Q`

contains the quantiles for the corresponding column in `A`

. `2`

, `8`

, and 10 are the quantiles of the first column of `A`

with elements 9, 10, 2, 10, 7, and 1. `Q = quantile(A,3)`

returns the same result because the default value of `dim`

is 1.

Calculate the quantiles for each row of `A`

for three evenly spaced cumulative probabilities.

Q = quantile(A,3,2)

`Q = `*6×3*
7.0000 8.0000 8.7500
4.2500 6.0000 9.5000
4.0000 8.0000 9.7500
1.0000 2.0000 8.5000
2.7500 5.0000 7.0000
2.5000 9.0000 10.0000

Each row of matrix `Q`

contains the three evenly spaced quantiles for the corresponding row in `A`

. `7`

, `8`

, and `8.75`

are the quantiles of the first row of `A`

with elements 9, 3, 10, 8, 7, 8, and 7.

### Quantiles of Multidimensional Array for Given Probabilities

Calculate the quantiles of a multidimensional array for specified probabilities by using `"all"`

and the `vecdim`

inputs.

Create a 3-by-5-by-2 array. Specify a vector of probabilities.

A = reshape(1:30,[3 5 2])

A = A(:,:,1) = 1 4 7 10 13 2 5 8 11 14 3 6 9 12 15 A(:,:,2) = 16 19 22 25 28 17 20 23 26 29 18 21 24 27 30

p = [0.25 0.75];

Calculate the 0.25 and 0.75 quantiles of all the elements of `A`

.

`Qall = quantile(A,p,"all")`

`Qall = `*2×1*
8
23

`Qall(1)`

is the 0.25 quantile of `A`

, and `Qall(2)`

is the 0.75 quantile of `A`

.

Calculate the 0.25 and 0.75 quantiles for each page of `A`

by specifying dimensions 1 and 2 as the operating dimensions.

Qpage = quantile(A,p,[1 2])

Qpage = Qpage(:,:,1) = 4.2500 11.7500 Qpage(:,:,2) = 19.2500 26.7500

`Qpage(1,1,1)`

is the 0.25 quantile of the first page of `A`

, and `Qpage(2,1,1)`

is the 0.75 quantile of the first page of `A`

.

Calculate the 0.25 and 0.75 quantiles of the elements in each `A(i,:,:)`

slice by specifying dimensions 2 and 3 as the operating dimensions.

Qrow = quantile(A,p,[2 3])

`Qrow = `*3×2*
7 22
8 23
9 24

`Qrow(3,1)`

is the 0.25 quantile of the elements in `A(3,:,:)`

, and `Qrow(3,2)`

is the 0.75 quantile of the elements in `A(3,:,:)`

.

### Median and Quartiles for Even Number of Data Elements

Find the median and quartiles of a vector with an even number of elements.

Create a data vector.

A = [2 5 6 10 11 13]

`A = `*1×6*
2 5 6 10 11 13

Calculate the median of the elements of `A`

.

Q = quantile(A,0.5)

Q = 8

Calculate the quartiles of the elements of `A`

.

Q = quantile(A,[0.25, 0.5, 0.75])

`Q = `*1×3*
5 8 11

Using `Q = quantile(A,3)`

is another way to compute the quartiles of the elements of `A`

.

These results might be different from the textbook definitions because `quantile`

uses Linear Interpolation to find the median and quartiles.

### Median and Quartiles for Odd Number of Data Elements

Find the median and quartiles of a vector with an odd number of elements.

Create a data vector.

A = [2 4 6 8 10 12 14]

`A = `*1×7*
2 4 6 8 10 12 14

Calculate the median of the elements of `A`

.

Q = quantile(A,0.50)

Q = 8

Calculate the quartiles of the elements of `A`

.

Q = quantile(A,[0.25, 0.5, 0.75])

`Q = `*1×3*
4.5000 8.0000 11.5000

Using `Q = quantile(A,3)`

is another way to compute the quartiles of `A`

.

These results might be different from the textbook definitions because `quantile`

uses Linear Interpolation to find the median and quartiles.

### Quantiles of Tall Vector for Given Probability

Calculate exact and approximate quantiles of a tall column vector for a given probability.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer`

function.

mapreducer(0)

Create a datastore for the `airlinesmall`

data set. Treat `"NA"`

values as missing data so that `datastore`

replaces them with `NaN`

values. Specify to work with the `ArrTime`

variable.

ds = datastore("airlinesmall.csv","TreatAsMissing","NA", ... "SelectedVariableNames","ArrTime");

Create a tall table `tt`

on top of the datastore, and extract the data from the tall table into a tall vector `A`

.

tt = tall(ds)

tt = Mx1 tall table ArrTime _______ 735 1124 2218 1431 746 1547 1052 1134 : :

A = tt{:,:}

A = Mx1 tall double column vector 735 1124 2218 1431 746 1547 1052 1134 : :

Calculate the exact quantile of `A`

for cumulative probability `p = 0.5`

. Because `A`

is a tall column vector and `p`

is a scalar, `quantile`

returns the exact quantile value by default.

p = 0.5; Qexact = quantile(A,p)

Qexact = tall double ?

Calculate the approximate quantile of `A`

for `p = 0.5`

. Specify `method`

as `"approximate"`

to use an approximation algorithm based on T-Digest for computing the quantiles.

Qapprox = quantile(A,p,"Method","approximate")

Qapprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :

Evaluate the tall arrays and bring the results into memory by using `gather`

.

[Qexact,Qapprox] = gather(Qexact,Qapprox)

Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 4: Completed in 0.81 sec - Pass 2 of 4: Completed in 0.26 sec - Pass 3 of 4: Completed in 0.45 sec - Pass 4 of 4: Completed in 0.31 sec Evaluation completed in 2.3 sec

Qexact = 1522

Qapprox = 1.5220e+03

The values of the exact quantile and the approximate quantile are the same to the four digits shown.

### Quantiles of Tall Matrix Along Different Dimensions

Calculate exact and approximate quantiles of a tall matrix for specified cumulative probabilities along different dimensions.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer`

function.

mapreducer(0)

Create a tall matrix `A`

containing a subset of variables stored in `varnames`

from the `airlinesmall`

data set. See Quantiles of Tall Vector for Given Probability for details about the steps to extract data from a tall array.

varnames = ["ArrDelay","ArrTime","DepTime","ActualElapsedTime"]; ds = datastore("airlinesmall.csv","TreatAsMissing","NA", ... "SelectedVariableNames",varnames); tt = tall(ds); A = tt{:,varnames}

A = Mx4 tall double matrix 8 735 642 53 8 1124 1021 63 21 2218 2055 83 13 1431 1332 59 4 746 629 77 59 1547 1446 61 3 1052 928 84 11 1134 859 155 : : : : : : : :

When operating along a dimension that is not 1, the `quantile`

function calculates the exact quantiles only so that it can perform the computation efficiently using a sorting-based algorithm (see Algorithms) instead of an approximation algorithm based on T-Digest.

Calculate the exact quantiles of `A`

along the second dimension for the vector `p`

of cumulative probabilities 0.25, 0.5, and 0.75.

p = [0.25 0.5 0.75]; Qexact = quantile(A,p,2)

Qexact = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :

When the function operates along the first dimension and `p`

is a vector of cumulative probabilities, you must use the approximation algorithm based on t-digest to compute the quantiles. Using the sorting-based algorithm to find quantiles along the first dimension of a tall array is computationally intensive.

Calculate the approximate quantiles of `A`

along the first dimension for the cumulative probabilities 0.25, 0.5, and 0.75. Because the default dimension is 1, you do not need to specify a value for `dim`

.

Qapprox = quantile(A,p,"Method","approximate")

Qapprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :

Evaluate the tall arrays and bring the results into memory by using `gather`

.

[Qexact,Qapprox] = gather(Qexact,Qapprox);

Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 1.7 sec Evaluation completed in 2.3 sec

Show the first five rows of the exact quantiles of `A`

(along the second dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.

Qexact(1:5,:)

ans =5×310^{3}× 0.0305 0.3475 0.6885 0.0355 0.5420 1.0725 0.0520 1.0690 2.1365 0.0360 0.6955 1.3815 0.0405 0.3530 0.6875

Each row of the matrix `Qexact`

contains the three quantiles of the corresponding row in `A`

. For example, `30.5`

, `347.5`

, and `688.5`

are the 0.25, 0.5, and 0.75 quantiles, respectively, of the first row in `A`

.

Show the approximate quantiles of `A`

(along the first dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.

Qapprox

Qapprox =3×410^{3}× -0.0070 1.1150 0.9322 0.0700 0 1.5220 1.3350 0.1020 0.0110 1.9180 1.7400 0.1510

Each column of the matrix `Qapprox`

contains to the three quantiles of the corresponding column in `A`

. For example, the first column of `Qapprox`

with elements –7, 0, and 11 contains the quantiles for the first column of `A`

.

### Quantiles of Tall Matrix for `n`

Evenly Spaced Probabilities

Calculate exact and approximate quantiles along different dimensions of a tall matrix for a given number of evenly spaced cumulative probabilities.

When you perform calculations on tall arrays, MATLAB® uses either a parallel pool (default if you have Parallel Computing Toolbox™) or the local MATLAB session. To run the example using the local MATLAB session when you have Parallel Computing Toolbox, change the global execution environment by using the `mapreducer`

function.

mapreducer(0)

Create a tall matrix `A`

containing a subset of variables stored in `varnames`

from the `airlinesmall`

data set. See Quantiles of Tall Vector for Given Probability for details about the steps to extract data from a tall array.

varnames = ["ArrDelay","ArrTime","DepTime","ActualElapsedTime"]; ds = datastore("airlinesmall.csv","TreatAsMissing","NA", ... "SelectedVariableNames",varnames); tt = tall(ds); A = tt{:,varnames}

A = Mx4 tall double matrix 8 735 642 53 8 1124 1021 63 21 2218 2055 83 13 1431 1332 59 4 746 629 77 59 1547 1446 61 3 1052 928 84 11 1134 859 155 : : : : : : : :

To calculate quantiles for evenly spaced cumulative probabilities along the first dimension, you must use the approximation algorithm based on T-Digest. Using the sorting-based algorithm (see Algorithms) to find quantiles along the first dimension of a tall array is computationally intensive.

Calculate the quantiles for three evenly spaced cumulative probabilities along the first dimension of `A`

. Because the default dimension is 1, you do not need to specify a value for `dim`

. Specify the `method`

as `"approximate"`

to use the approximation algorithm.

Qapprox = quantile(A,3,"Method","approximate")

Qapprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :

To calculate quantiles for evenly spaced cumulative probabilities along any other dimension (`dim`

is not `1`

), `quantile`

calculates the exact quantiles only, so that it can perform the computation efficiently by using the sorting-based algorithm.

Calculate the quantiles for three evenly spaced cumulative probabilities along the second dimension of `A`

. Because `dim`

is not 1, `quantile`

returns the exact quantiles by default.

Qexact = quantile(A,3,2)

Qexact = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : :

Evaluate the tall arrays and bring the results into memory by using `gather`

.

[Qapprox,Qexact] = gather(Qapprox,Qexact);

Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 1.3 sec Evaluation completed in 1.5 sec

Show the approximate quantiles of `A`

(along the first dimension) for the three evenly spaced cumulative probabilities.

Qapprox

Qapprox =3×410^{3}× -0.0070 1.1149 0.9321 0.0700 0 1.5220 1.3350 0.1020 0.0110 1.9180 1.7400 0.1510

Each column of the matrix `Qapprox`

contains the quantiles of the corresponding column in `A`

. For example, the first column of `Qapprox`

with elements –7, 0, and 11 contains the quantiles for the first column of `A`

.

Show the first five rows of the exact quantiles of `A`

(along the second dimension) for the three evenly spaced cumulative probabilities.

Qexact(1:5,:)

ans =5×310^{3}× 0.0305 0.3475 0.6885 0.0355 0.5420 1.0725 0.0520 1.0690 2.1365 0.0360 0.6955 1.3815 0.0405 0.3530 0.6875

Each row of the matrix `Qexact`

contains the three evenly spaced quantiles of the corresponding row in `A`

. For example, `30.5`

, `347.5`

, and `688.5`

are the 0.25, 0.5, and 0.75 quantiles, respectively, of the first row in `A`

.

## Input Arguments

`A`

— Input array

vector | matrix | multidimensional array

Input array, specified as a vector, matrix, or multidimensional array.

**Data Types: **`double`

| `single`

| `duration`

`p`

— Cumulative probabilities for which to compute quantiles

scalar | vector

Cumulative probabilities for which to compute quantiles, specified as a scalar or vector of scalars from 0 to 1.

**Example: **0.3

**Example: **[0.25, 0.5, 0.75]

**Example: **(0:0.25:1)

**Data Types: **`double`

| `single`

`n`

— Number of probabilities for which to compute quantiles

positive integer scalar

Number of probabilities for which to compute quantiles, specified as a positive
integer scalar. `quantile`

returns `n`

quantiles that
divide the data set into evenly distributed `n`

+1 segments.

**Data Types: **`double`

| `single`

`dim`

— Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify the dimension, then the default is the first array dimension of size greater than 1.

Consider an input matrix `A`

and a vector of cumulative
probabilities `p`

:

`Q = quantile(A,p,1)`

computes quantiles of the columns in`A`

for the cumulative probabilities in`p`

. Because 1 is the specified operating dimension,`Q`

has`length(p)`

rows.`Q = quantile(A,p,2)`

computes quantiles of the rows in`A`

for the cumulative probabilities in`p`

. Because 2 is the specified operating dimension,`Q`

has`length(p)`

columns.

Consider an input matrix `A`

and a vector of evenly spaced
probabilities `n`

:

`Q = quantile(A,n,1)`

computes quantiles of the columns in`A`

for the`n`

evenly spaced cumulative probabilities. Because 1 is the specified operating dimension,`Q`

has`n`

rows.`Q = quantile(A,n,2)`

computes quantiles of the rows in`A`

for the`n`

evenly spaced cumulative probabilities. Because 2 is the specified operating dimension,`Q`

has`n`

columns.

Dimension `dim`

indicates the dimension of `Q`

whose length is equal to `length(p)`

or `n`

.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`vecdim`

— Vector of dimensions to operate along

vector of positive integers

Vector of dimensions to operate along, specified as a vector of positive integers. Each element represents a dimension of the input data.

The size of the output `Q`

in the smallest specified operating
dimension is equal to `length(p)`

or `n`

. The size of
`Q`

in the other operating dimensions specified in
`vecdim`

is 1. The size of `Q`

in all dimensions not
specified in `vecdim`

remains the same as the input data.

Consider a 2-by-3-by-3 input array `A`

and the cumulative
probabilities `p`

. `quantile(A,p,[1 2])`

returns a
`length(p)`

-by-1-by-3 array because 1 and 2 are the operating
dimensions and `min([1 2]) = 1`

. Each page of the returned array
contains the quantiles of the elements on the corresponding page of
`A`

.

**Data Types: **`double`

| `single`

| `int8`

| `int16`

| `int32`

| `int64`

| `uint8`

| `uint16`

| `uint32`

| `uint64`

`method`

— Method for calculating quantiles

`"exact"`

(default) | `"approximate"`

## More About

### Linear Interpolation

Linear interpolation uses linear polynomials to find
*y _{i}* =
f(

*x*), the values of the underlying function

_{i}*Y*= f(

*X*) at the points in the vector or array

*x*. Given the data points (

*x*

_{1},

*y*

_{1}) and (

*x*

_{2},

*y*

_{2}), where

*y*

_{1}= f(

*x*

_{1}) and

*y*

_{2}= f(

*x*

_{2}), linear interpolation finds

*y*= f(

*x*) for a given

*x*between

*x*

_{1}and

*x*

_{2}as

$$y=f(x)={y}_{1}+\frac{\left(x-{x}_{1}\right)}{\left({x}_{2}-{x}_{1}\right)}\left({y}_{2}-{y}_{1}\right).$$

Similarly, if the 1.5/*n* quantile is *y*_{1.5/n} and
the 2.5/*n* quantile is *y*_{2.5/n},
then linear interpolation finds the 2.3/*n* quantile *y*_{2.3/n} as

$${y}_{\frac{2.3}{n}}={y}_{\frac{1.5}{n}}+\frac{\left(\frac{2.3}{n}-\frac{1.5}{n}\right)}{\left(\frac{2.5}{n}-\frac{1.5}{n}\right)}\left({y}_{\frac{2.5}{n}}-{y}_{\frac{1.5}{n}}\right).$$

### T-Digest

T-digest [2] is a probabilistic data structure that is a sparse representation of the empirical cumulative distribution function (CDF) of a data set. T-digest is useful for computing approximations of rank-based statistics (such as percentiles and quantiles) from online or distributed data in a way that allows for controllable accuracy, particularly near the tails of the data distribution.

For data that is distributed in different partitions, t-digest computes quantile
estimates (and percentile estimates) for each data partition separately, and then combines
the estimates while maintaining a constant-memory bound and constant relative accuracy of
computation ($$q(1-q)$$ for the *q*th quantile). For these reasons, t-digest is
practical for working with tall arrays.

To estimate quantiles of an array that is distributed in different partitions, first
build a t-digest in each partition of the data. A t-digest clusters the data in the
partition and summarizes each cluster by a centroid value and an accumulated weight that
represents the number of samples contributing to the cluster. T-digest uses large clusters
(widely spaced centroids) to represent areas of the CDF that are near

and uses small clusters (tightly spaced
centroids) to represent areas of the CDF that are near *q* = 0.5

and *q* =
0

.*q* = 1

T-digest controls the cluster size by using a scaling function that maps a quantile
*q* to an index *k* with a compression parameter
*δ*. That is,

$$k(q,\delta )=\delta \cdot \left(\frac{{\mathrm{sin}}^{-1}(2q-1)}{\pi}+\frac{1}{2}\right),$$

where the mapping *k* is monotonic with minimum value *k*(0,*δ*) = 0 and maximum value *k*(1,*δ*) =
*δ*. This figure shows the scaling function for *δ* = 10.

The scaling function translates the quantile *q* to the scaling factor
*k* in order to give variable-size steps in *q*. As a
result, cluster sizes are unequal (larger around the center quantiles and smaller near

and *q* = 0

). The smaller clusters allow for better accuracy near the edges of the
data.*q* =
1

To update a t-digest with a new observation that has a weight and location, find the cluster closest to the new observation. Then, add the weight and update the centroid of the cluster based on the weighted average, provided that the updated weight of the cluster does not exceed the size limitation.

You can combine independent t-digests from each partition of the data by taking a union of the t-digests and merging their centroids. To combine t-digests, first sort the clusters from all the independent t-digests in decreasing order of cluster weights. Then, merge neighboring clusters, when they meet the size limitation, to form a new t-digest.

Once you form a t-digest that represents the complete data set, you can estimate the endpoints (or boundaries) of each cluster in the t-digest and then use interpolation between the endpoints of each cluster to find accurate quantile estimates.

## Algorithms

For an *n*-element vector `A`

, `quantile`

computes quantiles by using a sorting-based algorithm:

The sorted elements in

`A`

are taken as the (0.5/*n*), (1.5/*n*), ..., ([*n*– 0.5]/*n*) quantiles. For example:For a data vector of five elements such as {6, 3, 2, 10, 1}, the sorted elements {1, 2, 3, 6, 10} respectively correspond to the 0.1, 0.3, 0.5, 0.7, and 0.9 quantiles.

For a data vector of six elements such as {6, 3, 2, 10, 8, 1}, the sorted elements {1, 2, 3, 6, 8, 10} respectively correspond to the (0.5/6), (1.5/6), (2.5/6), (3.5/6), (4.5/6), and (5.5/6) quantiles.

`quantile`

uses Linear Interpolation to compute quantiles for probabilities between (0.5/*n*) and ([*n*– 0.5]/*n*).For the quantiles corresponding to the probabilities outside that range,

`quantile`

assigns the minimum or maximum values of the elements in`A`

.

`quantile`

treats `NaN`

s
as missing values and removes them.

## References

[1] Langford, E. “Quartiles in
Elementary Statistics”, *Journal of Statistics Education*. Vol. 14, No.
3, 2006.

[2] Dunning, T., and O. Ertl. “Computing Extremely Accurate Quantiles Using T-Digests.” August 2017.

## Extended Capabilities

### Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

`Q = quantile(A,p)`

and`Q = quantile(A,n)`

return the exact quantiles (using a sorting-based algorithm) only if`A`

is a tall numeric column vector.`Q = quantile(__,dim)`

returns the exact quantiles only when*one*of these conditions exists:`A`

is a tall numeric column vector.`A`

is a tall numeric array and`dim`

is not`1`

. For example,`quantile(A,p,2)`

returns the exact quantiles along the rows of the tall array`A`

.

If

`A`

is a tall array and`dim`

is`1`

, then you must specify`method`

as`"approximate"`

to use an approximation algorithm based on T-Digest for computing the quantiles. For example,`quantile(A,p,1,"Method","approximate")`

returns the approximate quantiles along the columns of the tall array`A`

.`Q = quantile(__,vecdim)`

returns the exact quantiles only when*one*of these conditions exists:`A`

is a tall numeric column vector.`A`

is a tall numeric array and`vecdim`

does not include`1`

. For example, if`A`

is a 3-by-5-by-2 array, then`quantile(A,p,[2,3])`

returns the exact quantiles of the elements in each`A(i,:,:)`

slice.`A`

is a tall numeric array and`vecdim`

includes`1`

and all the dimensions of`A`

with a size greater than 1. For example, if`A`

is a 10-by-1-by-4 array, then`quantile(A,p,[1 3])`

returns the exact quantiles of the elements in`A(:,1,:)`

.

If

`A`

is a tall numeric array and`vecdim`

includes`1`

but does not include all the dimensions of`A`

with a size greater than 1, then you must specify`method`

as`"approximate"`

to use the approximation algorithm. For example, if`A`

is a 10-by-1-by-4 array, you can use`quantile(A,p,[1 2],"Method","approximate")`

to find the approximate quantiles of each page of`A`

.

For more information, see Tall Arrays.

### C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

The

`"all"`

and`vecdim`

inputs are not supported.The

`Method`

name-value argument is not supported.The

`dim`

input argument must be a compile-time constant.If you do not specify the

`dim`

input argument, the working (or operating) dimension can be different in the generated code. As a result, run-time errors can occur. For more details, see Automatic dimension restriction (MATLAB Coder).If the output

`Q`

is a vector, the orientation of`Q`

differs from MATLAB^{®}when all of these conditions are true:You do not supply

`dim`

.`A`

is a variable-size array, and not a variable-size vector, at compile time, but`A`

is a vector at run time.The orientation of the vector

`A`

does not match the orientation of the vector`p`

.

In this case, the output

`Q`

matches the orientation of`A`

, not the orientation of`p`

.

### GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

The

`"all"`

and`vecdim`

inputs are not supported.The

`Method`

name-value argument is not supported.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

## Version History

**Introduced before R2006a**

### R2022b: Improved performance with small input data

The `quantile`

function shows improved performance due to faster
input parsing. The performance improvement is most significant when input parsing is a
greater portion of the computation time. This situation occurs when:

The size of the input data is small.

The number of cumulative probabilities for which to compute quantiles is small.

Computation is along the default operating dimension.

For example, this code calculates four quantiles for a 3000-element matrix. The code is about 4.95x faster than in the previous release.

function timingQuantile A = rand(300,10); for k = 1:3e3 Q = quantile(A,[20 40 60 80]); end end

The approximate execution times are:

**R2022a:** 0.94 s

**R2022b:** 0.19 s

The code was timed on a Windows^{®} 10, Intel^{®}
Xeon^{®} CPU E5-1650 v4 @ 3.60 GHz test system using the `timeit`

function:

timeit(@timingQuantile)

### R2022a: Moved to MATLAB from Statistics and Machine Learning Toolbox

Previously, `quantile`

required Statistics and Machine Learning Toolbox™.

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