iqr

Interquartile range

Syntax

r = iqr(x)

r = iqr(x,'all')

r = iqr(x,dim)

r = iqr(x,vecdim)

r = iqr(pd)

Description

r = iqr(x) returns the interquartile range of the values in x.

• If x is a vector, then r is the difference between the 75th and the 25th percentiles of the data contained in x.

• If x is a matrix, then r is a row vector containing the difference between the 75th and the 25th percentiles of the sample data in each column of x.

• If x is a multidimensional array, then iqr operates along the first nonsingleton dimension of x. The size of this dimension becomes 1 while the sizes of all other dimensions remain the same.

r = iqr(x,'all') returns the interquartile range of all the values in x.

r = iqr(x,dim) returns the interquartile range along the dimension of x specified by dim.

r = iqr(x,vecdim) returns the interquartile range over the dimensions specified by vecdim. For example, if x is a matrix, then iqr(x,[1 2]) is the interquartile range of all the elements of x because every element of a matrix is contained in the array slice defined by dimensions 1 and 2.

r = iqr(pd) returns the interquartile range of the probability distribution pd.

Examples

Compute the Interquartile Range

Generate a 4-by-4 matrix of random data from a normal distribution with parameter values $$\mu$$ equal to 10 and $$\sigma$$ equal to 1.

rng default  % For reproducibility
x = normrnd(10,1,4)

x = 4×4

   10.5377   10.3188   13.5784   10.7254
   11.8339    8.6923   12.7694    9.9369
    7.7412    9.5664    8.6501   10.7147
   10.8622   10.3426   13.0349    9.7950

Compute the interquartile range for each column of data.

r = iqr(x)

r = 1×4

    2.2086    1.2013    2.5969    0.8541

Compute the interquartile range for each row of data.

r2 = iqr(x,2)

r2 = 4×1

    1.7237
    2.9870
    1.9449
    1.8797

Compute Interquartile Range of Multidimensional Array

Compute the interquartile range of a multidimensional array over multiple dimensions by specifying the 'all' and vecdim input arguments.

Create a 3-by-4-by-2 array X.

X = reshape(1:24,[3 4 2])

X = 
X(:,:,1) =

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


X(:,:,2) =

    13    16    19    22
    14    17    20    23
    15    18    21    24

Compute the interquartile range of all the values in X.

rall = iqr(X,'all')

rall = 12

Compute the interquartile range of each page of X. Specify the first and second dimensions as the operating dimensions along which the interquartile range is calculated.

rpage = iqr(X,[1 2])

rpage = 
rpage(:,:,1) =

     6


rpage(:,:,2) =

     6

For example, rpage(1,1,1) is the interquartile range of all the elements in X(:,:,1).

Compute the interquartile range of the elements in each X(i,:,:) slice by specifying the second and third dimensions as the operating dimensions.

rrow = iqr(X,[2 3])

rrow = 3×1

    12
    12
    12

For example, rrow(3) is the interquartile range of all the elements in X(3,:,:).

Compute the Normal Distribution Interquartile Range

Create a standard normal distribution object with the mean, $$\mu$$, equal to 0 and the standard deviation, $$\sigma$$, equal to 1.

pd = makedist('Normal','mu',0,'sigma',1);

Compute the interquartile range of the standard normal distribution.

r = iqr(pd)

r = 1.3490

The returned value is the difference between the 75th and the 25th percentile values for the distribution. This is equivalent to computing the difference between the inverse cumulative distribution function (icdf) values at the probabilities y equal to 0.75 and 0.25.

r2 = icdf(pd,0.75) - icdf(pd,0.25)

r2 = 1.3490

Interquartile Range of a Fitted Distribution

Load the sample data. Create a vector containing the first column of students’ exam grade data.

load examgrades;
x = grades(:,1);

Create a normal distribution object by fitting it to the data.

pd = fitdist(x,'Normal')

pd = 
  NormalDistribution

  Normal distribution
       mu = 75.0083   [73.4321, 76.5846]
    sigma =  8.7202   [7.7391, 9.98843]

Compute the interquartile range of the fitted distribution.

r = iqr(pd)

r = 11.7634

The returned result indicates that the difference between the 75th and 25th percentile of the students’ grades is 11.7634.

Use icdf to determine the 75th and 25th percentiles of the students’ grades.

y = icdf(pd,[0.25,0.75])

y = 1×2

   69.1266   80.8900

Calculate the difference between the 75th and 25th percentiles. This yields the same result as iqr.

y(2)-y(1)

ans = 11.7634

Use boxplot to visualize the interquartile range.

boxplot(x)

The top line of the box shows the 75th percentile, and the bottom line shows the 25th percentile. The center line shows the median, which is the 50th percentile.

Input Arguments

x — Input array

vector | matrix | multidimensional array

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

Data Types: single | double

dim — Dimension

positive integer value

Dimension along which the interquartile range is calculated, specified as a positive integer. For example, for a matrix x, when dim is equal to 1, iqr returns the interquartile range for the columns of x. When dim is equal to 2, iqr returns the interquartile range for the rows of x. For n-dimensional arrays, iqr operates along the first nonsingleton dimension of x.

Data Types: single | double

vecdim — Vector of dimensions

positive integer vector

Vector of dimensions, specified as a positive integer vector. Each element of vecdim represents a dimension of the input array x. The output r has length 1 in the specified operating dimensions. The other dimension lengths are the same for x and r.

For example, if x is a 2-by-3-by-3 array, then iqr(x,[1 2]) returns a 1-by-1-by-3 array. Each element of the output array is the interquartile range of the elements on the corresponding page of x.

Data Types: single | double

pd — Probability distribution

probability distribution object

Probability distribution, specified as one of the probability distribution objects in this table.

Distribution Object	Function or App to Create Probability Distribution Object
BetaDistribution	makedist, fitdist, Distribution Fitter
BinomialDistribution	makedist, fitdist, Distribution Fitter
BirnbaumSaundersDistribution	makedist, fitdist, Distribution Fitter
BurrDistribution	makedist, fitdist, Distribution Fitter
ExponentialDistribution	makedist, fitdist, Distribution Fitter
ExtremeValueDistribution	makedist, fitdist, Distribution Fitter
GammaDistribution	makedist, fitdist, Distribution Fitter
GeneralizedExtremeValueDistribution	makedist, fitdist, Distribution Fitter
GeneralizedParetoDistribution	makedist, fitdist, Distribution Fitter
HalfNormalDistribution	makedist, fitdist, Distribution Fitter
InverseGaussianDistribution	makedist, fitdist, Distribution Fitter
KernelDistribution	fitdist, Distribution Fitter
LogisticDistribution	makedist, fitdist, Distribution Fitter
LoglogisticDistribution	makedist, fitdist, Distribution Fitter
LognormalDistribution	makedist, fitdist, Distribution Fitter
LoguniformDistribution	makedist
MultinomialDistribution	makedist
NakagamiDistribution	makedist, fitdist, Distribution Fitter
NegativeBinomialDistribution	makedist, fitdist, Distribution Fitter
NormalDistribution	makedist, fitdist, Distribution Fitter
PiecewiseLinearDistribution	makedist
PoissonDistribution	makedist, fitdist, Distribution Fitter
RayleighDistribution	makedist, fitdist, Distribution Fitter
RicianDistribution	makedist, fitdist, Distribution Fitter
StableDistribution	makedist, fitdist, Distribution Fitter
tLocationScaleDistribution	makedist, fitdist, Distribution Fitter
TriangularDistribution	makedist
UniformDistribution	makedist
WeibullDistribution	makedist, fitdist, Distribution Fitter

Output Arguments

r — Interquartile range values

scalar | vector | matrix | multidimensional array

Interquartile range values, returned as a scalar, vector, matrix, or multidimensional array.

• If you input an array x, then the dimensions of r depend on whether the 'all', dim, or vecdim input arguments are specified. Each interquartile range value in r is the difference between the 75th and the 25th percentiles of the specified data contained in x.

• If you input a probability distribution pd, then the scalar value of r is the difference between the values of the 75th and 25th percentiles of the probability distribution.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• These input arguments are not supported: 'all' and vecdim.

• 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).

• The input argument pd can be a fitted probability distribution object for beta, exponential, extreme value, lognormal, normal, and Weibull distributions. Create pd by fitting a probability distribution to sample data from the fitdist function. For an example, see Code Generation for Probability Distribution Objects.

For more information on code generation, see Introduction to Code Generation and General Code Generation Workflow.

GPU Arrays

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

Usage notes and limitations:

• The input arguments 'all' and vecdim are not supported.

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