Documentation

quantile

Quantiles of a data set

Description

example

Y = quantile(X,p) returns quantiles of the elements in data vector or array X for the cumulative probability or probabilities p in the interval [0,1].

• If X is a vector, then Y is a scalar or a vector having the same length as p.

• If X is a matrix, then Y is a row vector or a matrix where the number of rows of Y is equal to the length of p.

• For multidimensional arrays, quantile operates along the first nonsingleton dimension of X.

example

Y = quantile(X,N) returns quantiles for N evenly spaced cumulative probabilities (1/(N + 1), 2/(N + 1), ..., N/(N + 1)) for integer N>1.

• If X is a vector, then Y is a scalar or a vector with length N.

• If X is a matrix, then Y is a matrix where the number of rows of Y is equal to N.

• For multidimensional arrays, quantile operates along the first nonsingleton dimension of X.

example

Y = quantile(___,'all') returns quantiles of all the elements of X for either of the first two syntaxes.

example

Y = quantile(___,dim) returns quantiles along the operating dimension dim for either of the first two syntaxes.

example

Y = quantile(___,vecdim) returns quantiles over the dimensions specified in the vector vecdim for either of the first two syntaxes. For example, if X is a matrix, then quantile(X,0.5,[1 2]) returns the 0.5 quantile of all the elements of X because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.

example

Y = quantile(___,'Method',method) returns either exact or approximate quantiles based on the value of method, using any of the input argument combinations in the previous syntaxes.

Examples

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Calculate the quantiles of a data set for specified probabilities.

Generate a data set of size 10.

rng('default'); % for reproducibility
x = normrnd(0,1,1,10)
x = 1×10

0.5377    1.8339   -2.2588    0.8622    0.3188   -1.3077   -0.4336    0.3426    3.5784    2.7694

Calculate the 0.3 quantile.

y = quantile(x,0.30)
y = -0.0574

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

y = quantile(x,[0.025 0.25 0.50 0.75 0.975])
y = 1×5

-2.2588   -0.4336    0.4401    1.8339    3.5784

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

Generate a data set of size 10.

rng('default'); % for reproducibility
x = normrnd(0,1,1,10)
x = 1×10

0.5377    1.8339   -2.2588    0.8622    0.3188   -1.3077   -0.4336    0.3426    3.5784    2.7694

Calculate four evenly spaced quantiles.

y = quantile(x,4)
y = 1×4

-0.8706    0.3307    0.6999    2.3017

Using y = quantile(x,[0.2,0.4,0.6,0.8]) is another way to return the four evenly spaced quantiles.

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
X = normrnd(0,1,4,6)
X = 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 X (dim = 1).

y = quantile(X,0.3,1)
y = 1×6

-0.3013   -0.6958    1.5336   -0.1056    0.9491    0.1078

quantile returns a row vector y when calculating one quantile for each column of a matrix. For example, -0.3013 is the 0.3 quantile of the first column of X with elements (0.5377, 1.8339, -2.2588, 0.8622). Because the default value of dim is 1, you can return the same result with y = quantile(X,0.3).

Calculate the 0.3 quantile for each row of X (dim = 2).

y = quantile(X,0.3,2)
y = 4×1

0.3844
-0.8642
-1.0750
0.4985

quantile returns a column vector y when calculating one quantile for each row of a matrix. For example 0.3844 is the 0.3 quantile of the first row of X with elements (0.5377, 0.3188, 3.5784, 0.7254, -0.1241, 0.6715).

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

Generate a 6-by-10 data matrix.

rng('default');  % for reproducibility
X = unidrnd(10,6,7)
X = 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 three evenly spaced quantiles for each column of X (dim = 1).

y = quantile(X,3,1)
y = 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 y corresponds to the three evenly spaced quantiles of each column of matrix X. For example, the first column of y with elements (2, 8, 10) has the quantiles for the first column of X with elements (9, 10, 2, 10, 7, 1). y = quantile(X,3) returns the same answer because the default value of dim is 1.

Calculate three evenly spaced quantiles for each row of X (dim = 2).

y = quantile(X,3,2)
y = 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 y corresponds to the three evenly spaced quantiles of each row of matrix X. For example, the first row of y with elements (7, 8, 8.75) has the quantiles for the first row of X with elements (9, 3, 10, 8, 7, 8, 7).

Calculate the quantiles of a multidimensional array for specified probabilities by using the 'all' and vecdim input arguments.

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

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

1     4     7    10    13
2     5     8    11    14
3     6     9    12    15

X(:,:,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 in X.

Yall = quantile(X,p,'all')
Yall = 2×1

8
23

Yall(1) is the 0.25 quantile of X, and Yall(2) is the 0.75 quantile of X.

Calculate the 0.25 and 0.75 quantiles for each page of X by specifying dimensions 1 and 2 as the operating dimensions.

Ypage = quantile(X,p,[1 2])
Ypage =
Ypage(:,:,1) =

4.2500
11.7500

Ypage(:,:,2) =

19.2500
26.7500

For example, Ypage(1,1,1) is the 0.25 quantile of the first page of X, and Ypage(2,1,1) is the 0.75 quantile of the first page of X.

Calculate the 0.25 and 0.75 quantiles of the elements in each X(i,:,:) slice by specifying dimensions 2 and 3 as the operating dimensions.

Yrow = quantile(X,p,[2 3])
Yrow = 3×2

7    22
8    23
9    24

For example, Yrow(3,1) is the 0.25 quantile of the elements in X(3,:,:), and Yrow(3,2) is the 0.75 quantile of the elements in X(3,:,:).

Find median and quartiles of a vector, x, with even number of elements.

Enter the data.

x = [2 5 6 10 11 13]
x = 1×6

2     5     6    10    11    13

Calculate the median of x.

y = quantile(x,0.50)
y = 8

Calculate the quartiles of x.

y = quantile(x,[0.25, 0.5, 0.75])
y = 1×3

5     8    11

Using y = quantile(x,3) is another way to compute the quartiles of x.

These results might be different than the textbook definitions because quantile uses Linear Interpolation to find the median and quartiles.

Find median and quartiles of a vector, x, with odd number of elements.

Enter the data.

x = [2 4 6 8 10 12 14]
x = 1×7

2     4     6     8    10    12    14

Find the median of x.

y = quantile(x,0.50)
y = 8

Find the quartiles of x.

y = quantile(x,[0.25, 0.5, 0.75])
y = 1×3

4.5000    8.0000   11.5000

Using y = quantile(x,3) is another way to compute the quartiles of x.

These results might be different than the textbook definitions because quantile uses Linear Interpolation to find the median and quartiles.

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. If you want to run the example using the local MATLAB session when you have Parallel Computing Toolbox, you can change the global execution environment by using the mapreducer function.

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

t = tall(ds) % Tall table
Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 4).

t =

Mx1 tall table

ArrTime
_______

735
1124
2218
1431
746
1547
1052
1134
:
:
x = t{:,:}   % Tall vector
x =

Mx1 tall double column vector

735
1124
2218
1431
746
1547
1052
1134
:
:

Calculate the exact quantile of x for p = 0.5. Because X is a tall column vector and p is a scalar, quantile returns the exact quantile value by default.

p = 0.5; % Cumulative probability
yExact = quantile(x,p)
yExact =

tall double

?

Calculate the approximate quantile of x for p = 0.5. Specify 'Method','approximate' to use an approximation algorithm based on T-Digest for computing the quantiles.

yApprox = quantile(x,p,'Method','approximate')
yApprox =

MxNx... tall double array

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

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

[yExact,yApprox] = gather(yExact,yApprox)
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 4: Completed in 4.8 sec
- Pass 2 of 4: Completed in 1.2 sec
- Pass 3 of 4: Completed in 1.7 sec
- Pass 4 of 4: Completed in 1.2 sec
Evaluation completed in 12 sec
yExact = 1522
yApprox = 1.5220e+03

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

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. If you want to run the example using the local MATLAB session when you have Parallel Computing Toolbox, you can change the global execution environment by using the mapreducer function.

Create a tall matrix X containing a subset of variables 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'}; % Subset of variables in the data set
ds = datastore('airlinesmall.csv','TreatAsMissing','NA',...
'SelectedVariableNames',varnames); % Datastore
t = tall(ds);     % Tall table
Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 4).
X = t{:,varnames} % Tall matrix
X =

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

p = [0.25 0.50 0.75]; % Vector of cumulative probabilities
Yexact = quantile(X,p,2)
Yexact =

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 the quantiles along the first dimension of a tall array is computationally intensive.

Calculate the approximate quantiles of X 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.

Yapprox = quantile(X,p,'Method','approximate')
Yapprox =

MxNx... tall double array

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

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

[Yexact,Yapprox] = gather(Yexact,Yapprox);
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 7.4 sec
Evaluation completed in 9.2 sec

Show the first five rows of the exact quantiles of X (along the second dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.

Yexact(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 Yexact contains the three quantiles of the corresponding row in X. 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 X.

Show the approximate quantiles of X (along the first dimension) for the cumulative probabilities 0.25, 0.5, and 0.75.

Yapprox
Yapprox = 3×4
103 ×

-0.0070    1.1149    0.9322    0.0700
0    1.5220    1.3350    0.1020
0.0110    1.9180    1.7400    0.1510

Each column of the matrix Yapprox corresponds to the three quantiles for each column of the matrix X. For example, the first column of Yapprox with elements (–7, 0, 11) contains the quantiles for the first column of X.

Calculate exact and approximate quantiles along different dimensions of a tall matrix for N 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. If you want to run the example using the local MATLAB session when you have Parallel Computing Toolbox, you can change the global execution environment by using the mapreducer function.

Create a tall matrix X containing a subset of variables 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'}; % Subset of variables in the data set
ds = datastore('airlinesmall.csv','TreatAsMissing','NA',...
'SelectedVariableNames',varnames); % Datastore
t = tall(ds); % Tall table
Starting parallel pool (parpool) using the 'local' profile ...
Connected to the parallel pool (number of workers: 4).
X = t{:,varnames}
X =

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 find evenly spaced quantiles 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 three evenly spaced quantiles along the first dimension of X. Because the default dimension is 1, you do not need to specify a value for dim. Specify 'Method','approximate' to use the approximation algorithm.

N = 3; % Number of quantiles
Yapprox = quantile(X,N,'Method','approximate')
Yapprox =

MxNx... tall double array

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

To find evenly spaced quantiles 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 three evenly spaced quantiles along the second dimension of X. Because dim is not 1, quantile returns the exact quantiles by default.

Yexact = quantile(X,N,2)
Yexact =

MxNx... tall double array

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

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

[Yapprox,Yexact] = gather(Yapprox,Yexact);
Evaluating tall expression using the Parallel Pool 'local':
- Pass 1 of 1: Completed in 5.7 sec
Evaluation completed in 7.7 sec

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

Yapprox
Yapprox = 3×4
103 ×

-0.0070    1.1149    0.9322    0.0700
0    1.5220    1.3350    0.1020
0.0110    1.9180    1.7400    0.1510

Each column of the matrix Yapprox corresponds to the three evenly spaced quantiles for each column of the matrix X. For example, the first column of Yapprox with elements (–7, 0, 11) contains the quantiles for the first column of X.

Show the first five rows of the exact quantiles of X (along the second dimension) for the three evenly spaced cumulative probabilities.

Yexact(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 Yexact contains the three evenly spaced quantiles of the corresponding row in X. 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 X.

Input Arguments

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Input data, specified as a vector or array.

Data Types: double | single

Cumulative probabilities for which to compute the 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

Number of quantiles to compute, specified as a positive integer. quantile returns N quantiles that divide the data set into evenly distributed N+1 segments.

Data Types: double | single

Dimension along which the quantiles of a matrix X are requested, specified as a positive integer. For example, for a matrix X, when dim = 1, quantile returns the quantile(s) of the columns of X; when dim = 2, quantile returns the quantile(s) of the rows of X. For a multidimensional array X, the length of the dimth dimension of Y is the same as the length of p.

Data Types: single | double

Vector of dimensions, specified as a positive integer vector. Each element of vecdim represents a dimension of the input array X. In the smallest specified operating dimension (that is, dimension min(vecdim)), the output Y has length equal to the number of quantiles requested (either N or length(p)). In each of the remaining operating dimensions, Y has length 1. The other dimension lengths are the same for X and Y.

For example, consider a 2-by-3-by-3 array X with p = [0.2 0.4 0.6 0.8]. In this case, quantile(X,p,[1 2]) returns an array, where each page of the array contains the 0.2, 0.4, 0.6, and 0.8 quantiles of the elements on the corresponding page of X. Because 1 and 2 are the operating dimensions, with min([1 2]) = 1 and length(p) = 4, the output is a 4-by-1-by-3 array.

Data Types: single | double

Method for calculating quantiles, specified as 'exact' or 'approximate'. By default, quantile returns the exact quantiles by implementing an algorithm that uses sorting. You can specify 'method','approximate' for quantile to return approximate quantiles by implementing an algorithm that uses T-Digest.

Data Types: char | string

Output Arguments

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Quantiles of a data vector or array, returned as a scalar or array for one or multiple values of cumulative probabilities.

• If X is a vector, then Y is a scalar or a vector with the same length as the number of quantiles requested (N or length(p)). Y(i) contains the p(i) quantile.

• If X is an array of dimension d, then Y is an array with the length of the smallest operating dimension equal to the number of quantiles requested (N or length(p)).

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Multidimensional Array

A multidimensional array is an array with more than two dimensions. For example, if X is a 1-by-3-by-4 array, then X is a 3-D array.

Nonsingleton Dimension

A nonsingleton dimension of an array is a dimension whose size is not equal to 1. A first nonsingleton dimension of an array is the first dimension that satisfies the nonsingleton condition. For example, if X is a 1-by-1-by-2-by-4 array, then the third dimension is the first nonsingleton dimension of X.

Linear Interpolation

Linear interpolation uses linear polynomials to find yi = f(xi), 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 follows:

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

Similarly, if the 1.5/n quantile is y1.5/n and the 2.5/n quantile is y2.5/n, then linear interpolation finds the 2.3/n quantile y2.3/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 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 ($q\left(1-q\right)$ 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 or 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 $\delta$. That is,

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

where the mapping k is monotonic with minimum value k(0,δ) = 0 and maximum value k(1,δ) = δ. The following 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 or 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 end-points (or boundaries) of each cluster in the t-digest and then use interpolation between the end-points of each cluster to find accurate quantile estimates.

Algorithms

For an n-element vector X, quantile computes quantiles by using a sorting-based algorithm as follows:

1. The sorted elements in X 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, 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), (5.5/6) quantiles.

2. quantile uses Linear Interpolation to compute quantiles for probabilities between (0.5/n) and ([n – 0.5]/n).

3. For the quantiles corresponding to the probabilities outside that range, quantile assigns the minimum or maximum values of the elements in X.

quantile treats NaNs as missing values and removes them.

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

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