# prctile

Percentiles of data set

## Syntax

``P = prctile(A,p)``
``P = prctile(A,p,"all")``
``P = prctile(A,p,dim)``
``P = prctile(A,p,vecdim)``
``P = prctile(___,"Method",method)``

## Description

example

````P = prctile(A,p)` returns percentiles of elements in input data `A` for the percentages `p` in the interval [0,100]. If `A` is a vector, then `P` is a scalar or a vector with the same length as `p`. `P(i)` contains the `p(i)` percentile.If `A` is a matrix, then `P` is a row vector or a matrix, where the number of rows of `P` is equal to `length(p)`. The `i`th row of `P` contains the `p(i)` percentiles of each column of `A`.If `A` is a multidimensional array, then `P` contains the percentiles computed along the first array dimension of size greater than 1. ```

example

````P = prctile(A,p,"all")` returns percentiles of all the elements in `x`.```

example

````P = prctile(A,p,dim)` operates along the dimension `dim`. For example, if `A` is a matrix, then `prctile(A,p,2)` operates on the elements in each row.```

example

````P = prctile(A,p,vecdim)` operates along the dimensions specified in the vector `vecdim`. For example, if `A` is a matrix, then `prctile(A,p,[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.```

example

````P = prctile(___,"Method",method)` returns either exact or approximate percentiles based on the value of `method`, using any of the input argument combinations in the previous syntaxes.```

## Examples

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Calculate the percentile of a data set for a given percentage.

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 42nd percentile of the elements of `A`.

`P = prctile(A,42)`
```P = -0.1026 ```

Find the percentiles of all the values in an array.

Create a 3-by-5-by-2 array.

```rng default % for reproducibility A = randn(3,5,2)```
```A = A(:,:,1) = 0.5377 0.8622 -0.4336 2.7694 0.7254 1.8339 0.3188 0.3426 -1.3499 -0.0631 -2.2588 -1.3077 3.5784 3.0349 0.7147 A(:,:,2) = -0.2050 1.4090 -1.2075 0.4889 -0.3034 -0.1241 1.4172 0.7172 1.0347 0.2939 1.4897 0.6715 1.6302 0.7269 -0.7873 ```

Find the 40th and 60th percentiles of all the elements of `A`.

`P = prctile(A,[40 60],"all")`
```P = 2×1 0.3307 0.7213 ```

`P(1)` is the 40th percentile of `A`, and `P(2)` is the 60th percentile of `A`.

Calculate the percentiles along the columns and rows of a data matrix for specified percentages.

Generate a 5-by-5 data matrix.

`A = (1:5)'*(2:6)`
```A = 5×5 2 3 4 5 6 4 6 8 10 12 6 9 12 15 18 8 12 16 20 24 10 15 20 25 30 ```

Calculate the 25th, 50th, and 75th percentiles for each column of `A`.

`P = prctile(A,[25 50 75],1)`
```P = 3×5 3.5000 5.2500 7.0000 8.7500 10.5000 6.0000 9.0000 12.0000 15.0000 18.0000 8.5000 12.7500 17.0000 21.2500 25.5000 ```

Each column of matrix `P` contains the three percentiles for the corresponding column in matrix `A`. `7`, `12`, and `17` are the 25th, 50th, and 75th percentiles of the third column of `A` with elements 4, 8, 12, 16, and 20. `P = prctile(A,[25 50 75])` returns the same result.

Calculate the 25th, 50th, and 75th percentiles along the rows of `A`.

`P = prctile(A,[25 50 75],2)`
```P = 5×3 2.7500 4.0000 5.2500 5.5000 8.0000 10.5000 8.2500 12.0000 15.7500 11.0000 16.0000 21.0000 13.7500 20.0000 26.2500 ```

Each row of matrix `P` contains the three percentiles for the corresponding row in matrix `A`. `2.75`, `4`, and `5.25` are the 25th, 50th, and 75th percentiles of the first row of `A` with elements 2, 3, 4, 5, and 6.

Find the percentiles of a multidimensional array along multiple dimensions.

Create a 3-by-5-by-2 array.

`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 ```

Calculate the 40th and 60th percentiles for each page of `A` by specifying dimensions 1 and 2 as the operating dimensions.

`Ppage = prctile(A,[40 60],[1 2])`
```Ppage = Ppage(:,:,1) = 6.5000 9.5000 Ppage(:,:,2) = 21.5000 24.5000 ```

`Ppage(1,1,1)` is the 40th percentile of the first page of `A`, and `Ppage(2,1,1)` is the 60th percentile of the first page of `A`.

Calculate the 40th and 60th percentiles of the elements in each `A(:,i,:)` slice by specifying dimensions 1 and 3 as the operating dimensions.

`Pcol = prctile(A,[40 60],[1 3])`
```Pcol = 2×5 2.9000 5.9000 8.9000 11.9000 14.9000 16.1000 19.1000 22.1000 25.1000 28.1000 ```

`Pcol(1,4)` is the 40th percentile of the elements in `A(:,4,:)`, and `Pcol(2,4)` is the 60th percentile of the elements in `A(:,4,:)`.

Calculate exact and approximate percentiles of a tall column vector for a given percentage.

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 50th percentile of `A`. Because `A` is a tall column vector and `p` is a scalar, `prctile` returns the exact percentile value by default.

```p = 50; Pexact = prctile(A,p)```
```Pexact = tall double ? ```

Calculate the approximate 50th percentile of `A`. Specify the `"approximate"` method to use an approximation algorithm based on T-Digest for computing the percentile.

`Papprox = prctile(A,p,"Method","approximate")`
```Papprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```

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

`[Pexact,Papprox] = gather(Pexact,Papprox)`
```Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 4: Completed in 0.69 sec - Pass 2 of 4: Completed in 0.25 sec - Pass 3 of 4: Completed in 0.38 sec - Pass 4 of 4: Completed in 0.32 sec Evaluation completed in 2.2 sec ```
```Pexact = 1522 ```
```Papprox = 1.5220e+03 ```

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

Calculate exact and approximate percentiles of a tall matrix for specified percentages 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 Percentiles of Tall Vector for Given Percentage 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 `prctile` function calculates exact percentiles only so that it can compute efficiently using a sorting-based algorithm (see Algorithms) instead of an approximation algorithm based on T-Digest.

Calculate the exact 25th, 50th, and 75th percentiles of `A` along the second dimension.

```p = [25 50 75]; Pexact = prctile(A,p,2)```
```Pexact = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```

When the function operates along the first dimension and `p` is a vector of percentages, you must use the approximation algorithm based on t-digest to compute the percentiles. Using the sorting-based algorithm to find percentiles along the first dimension of a tall array is computationally intensive.

Calculate the approximate 25th, 50th, and 75th percentiles of `A` along the first dimension. Because the default dimension is 1, you do not need to specify a value for `dim`.

`Papprox = prctile(A,p,"Method","approximate")`
```Papprox = MxNx... tall double array ? ? ? ... ? ? ? ... ? ? ? ... : : : : : : ```

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

`[Pexact,Papprox] = gather(Pexact,Papprox);`
```Evaluating tall expression using the Local MATLAB Session: - Pass 1 of 1: Completed in 1.5 sec Evaluation completed in 1.9 sec ```

Show the first five rows of the exact 25th, 50th, and 75th percentiles along the second dimension of `A`.

`Pexact(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 `Pexact` contains the three percentiles of the corresponding row in `A`. `30.5`, `347.5`, and `688.5` are the 25th, 50th, and 75th percentiles, respectively, of the first row in `A`.

Show the approximate 25th, 50th, and 75th percentiles of `A` along the first dimension.

`Papprox`
```Papprox = 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 `Papprox` contains the three percentiles of the corresponding column in `A`. The first column of `Papprox` contains the percentiles for the first column of `A`.

## Input Arguments

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

Data Types: `single` | `double`

Percentages for which to compute percentiles, specified as a scalar or vector of scalars from 0 to 100.

Example: 25

Example: [25, 50, 75]

Data Types: `single` | `double`

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 percentages `p`:

• `P = prctile(A,p,1)` computes percentiles of the columns in `A` for the percentages in `p`.

• `P = prctile(A,p,2)` computes percentiles of the rows in `A` for the percentages in `p`.

Dimension `dim` indicates the dimension of `P` that has the same length as `p`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`

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 `P` in the smallest specified operating dimension is equal to the length of `p`. The size of `P` in the other operating dimensions specified in `vecdim` is 1. The size of `P` 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 percentages `p`. `prctile(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 percentiles of the elements on the corresponding page of `A`.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`

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

• `"exact"` — Calculate exact percentiles with an algorithm that uses sorting.

• `"approximate"` — Calculate approximate percentiles with an algorithm that uses T-Digest.

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### 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

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

Similarly, if the 100(1.5/n)th percentile is y1.5/n and the 100(2.5/n)th percentile is y2.5/n, then linear interpolation finds the 100(2.3/n)th percentile, 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``` 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,

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

where the mapping k is monotonic with minimum value k(0,δ) = 0 and maximum value k(1,δ) = δ. 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`, `prctile` returns percentiles by using a sorting-based algorithm:

1. The sorted elements in `A` are taken as the 100(0.5/n)th, 100(1.5/n)th, ..., 100([n – 0.5]/n)th percentiles. 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 10th, 30th, 50th, 70th, and 90th percentiles.

• 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 (50/6)th, (150/6)th, (250/6)th, (350/6)th, (450/6)th, and (550/6)th percentiles.

2. `prctile` uses linear interpolation to compute percentiles for percentages between 100(0.5/n) and 100([n – 0.5]/n).

3. `prctile` assigns the minimum or maximum values of the elements in `A` to the percentiles corresponding to the percentages outside that range.

`prctile` treats `NaN`s 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.