quantile

Quantiles of data set

Syntax

Q = quantile(A,p)

Q = quantile(A,n)

Q = quantile(___,"all")

Q = quantile(___,dim)

Q = quantile(___,vecdim)

Q = quantile(___,Method=method)

Description

Q = quantile(A,p) returns quantiles of elements in input data 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 ith 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 whose size does not equal 1.

Q = quantile(A,n) returns quantiles for 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 whose size does not equal 1.

Q = quantile(___,"all") returns quantiles of all the elements of A for either of the first two syntaxes.

Q = quantile(___,dim) operates along the dimension 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.

Q = quantile(___,vecdim) operates along the dimensions specified in the vector 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.

Q = quantile(___,Method=method) calculates quantiles using the specified method. Specify the method in addition to any of the input argument combinations in the previous syntaxes.

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 =

  M×1 tall table

    ArrTime
    _______

      735  
     1124  
     2218  
     1431  
      746  
     1547  
     1052  
     1134  
       :
       :

A = tt{:,:}

A =

  M×1 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

    ?

Preview deferred. Learn more.

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 =

  M×N×... tall array

    ?    ?    ?    ...
    ?    ?    ?    ...
    ?    ?    ?    ...
    :    :    :
    :    :    :

Preview deferred. Learn more.

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 0.63 sec
- Pass 2 of 4: Completed in 0.23 sec
- Pass 3 of 4: Completed in 0.36 sec
- Pass 4 of 4: Completed in 0.26 sec
Evaluation completed in 1.9 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 =

  M×4 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 =

  M×N×... tall array

    ?    ?    ?    ...
    ?    ?    ?    ...
    ?    ?    ?    ...
    :    :    :
    :    :    :

Preview deferred. Learn more.

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 =

  M×N×... tall array

    ?    ?    ?    ...
    ?    ?    ?    ...
    ?    ?    ?    ...
    :    :    :
    :    :    :

Preview deferred. Learn more.

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 1.9 sec
Evaluation completed in 2.5 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 = 5×3
103 ×

    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 = 3×4
103 ×

   -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 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 =

  M×4 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 =

  M×N×... tall array

    ?    ?    ?    ...
    ?    ?    ?    ...
    ?    ?    ?    ...
    :    :    :
    :    :    :

Preview deferred. Learn more.

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 =

  M×N×... tall array

    ?    ?    ?    ...
    ?    ?    ?    ...
    ?    ?    ?    ...
    :    :    :
    :    :    :

Preview deferred. Learn more.

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 sec
Evaluation completed in 1.2 sec

Show the approximate quantiles of A (along the first dimension) for the three evenly spaced cumulative probabilities.

Qapprox

Qapprox = 3×4
103 ×

   -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 = 5×3
103 ×

    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 whose size does not equal 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

"midpoint" (default) | "inclusive" | "exclusive" | "approximate"

Method for calculating quantiles, specified as one of these values:

• "midpoint" — Calculate quantiles with a midpoint algorithm that uses sorting.

  Before R2025a: Use "exact" for this method.

• "inclusive" — Calculate quantiles with an algorithm that uses sorting and includes the 0th and 1st quantiles within the bounds of the data. (since R2025a)

• "exclusive" — Calculate quantiles with an algorithm that uses sorting and excludes the 0th and 1st quantiles from the bounds of the data. (since R2025a)

• "approximate" — Calculate approximate quantiles with an algorithm that uses T-Digest for a double or single input array.

For more information about the quantile calculations, see Algorithms.

More About

Linear Interpolation

Linear interpolation uses linear polynomials to find y_i = f(x_i), the values of the underlying function Y = f(X) at the points in the vector or array x. Given the data points (x₁, y₁) and (x₂, y₂), where y₁ = f(x₁) and y₂ = f(x₂), linear interpolation finds y = f(x) for a given x between x₁ and x₂ 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 qth 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 q = 0.5 and uses small clusters (tightly spaced centroids) to represent areas of the CDF that are near q = 0 and 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,

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

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 q = 0 and q = 1). The smaller clusters allow for better accuracy near the edges of the data.

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, the quantile function computes quantiles by using a sorting-based algorithm when you choose any method except "approximate".

1. The sorted elements in A are mapped to quantiles based on the method you choose, as described in this table.

   Quantile	Method
   	"midpoint" Before R2025a: "exact"	"inclusive" (since R2025a)	"exclusive" (since R2025a)
   Quantile of 1st sorted element	1/(2n)	0	1/(n+1)
   Quantile of 2nd sorted element	3/(2n)	1/(n−1)	2/(n+1)
   Quantile of 3rd sorted element	5/(2n)	2/(n−1)	3/(n+1)
   ...	...	...	...
   Quantile of kth sorted element	(2k−1)/(2n)	(k−1)/(n−1)	k/(n+1)
   ...	...	...	...
   Quantile of (n−1)th sorted element	(2n−3)/(2n)	(n−2)/(n−1)	(n−1)/(n+1)
   Quantile of nth sorted element	(2n−1)/(2n)	1	n/(n+1)

   For example, if A is [6 3 2 10 1], then the quantiles are as shown in this table.

   Quantile	Method
   	"midpoint" Before R2025a: "exact"	"inclusive" (since R2025a)	"exclusive" (since R2025a)
   Quantile of 1	1/10	0	1/6
   Quantile of 2	3/10	1/4	1/3
   Quantile of 3	1/2	1/2	1/2
   Quantile of 6	7/10	3/4	2/3
   Quantile of 10	9/10	1	5/6

2. The quantile function uses linear interpolation to compute quantiles for probabilities between that of the first and that of the last sorted element of A. For more information, see Linear Interpolation.

   For example, if A is [6 3 2 10 1], then:

   • For the midpoint method, the 0.4th quantile is 2.5.

     Before R2025a: For the exact method, the 0.4th quantile is 2.5.

   • For the inclusive method, the 0.4th quantile is 2.6. (since R2025a)

   • For the exclusive method, the 0.4th quantile is 2.4. (since R2025a)

3. The quantile function assigns the minimum or maximum values of the elements in A to the quantiles corresponding to the probabilities outside of that range.

   For example, if A is [6 3 2 10 1], then, for both the midpoint and exclusive method, the 0.05th quantile is 1. (since R2025a)

   Before R2025a: For example, if A is [6 3 2 10 1], then, for the exact method, the 0.05th quantile is 1.

The quantile function treats NaN values 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.

The quantile function supports tall arrays with the following 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 whose size does not equal 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 whose size does not equal 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 Incompatibility with MATLAB for Default Dimension Selection (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™.

The quantile function supports GPU array input with these 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

R2025a: Calculate quantiles using inclusive and exclusive methods

You can calculate quantiles using an inclusive or exclusive method. Specify the method name as "inclusive" or "exclusive", respectively. The "inclusive" method includes the 0th and 1st quantiles within the bounds of the data; the "exclusive" method excludes them.

Additionally, the name of the default method has changed from "exact" to "midpoint". The quantile function continues to support Method="exact" for backward compatibility.

These examples illustrate the differences between the default method and the two new methods.

x = 1:5; Q = quantile(x,1/4,Method="midpoint") Q = 1.7500

x = 1:5; Q = quantile(x,1/4,Method="inclusive") Q = 2

x = 1:5; Q = quantile(x,1/4,Method="exclusive") Q = 1.5000

R2022a: Moved to MATLAB from Statistics and Machine Learning Toolbox

Previously, quantile required Statistics and Machine Learning Toolbox™.