# InterquartileRange InterquartileRange[data]

gives the difference between the upper and lower quartiles for the elements in data.

InterquartileRange[data,{{a,b},{c,d}}]

uses the quantile definition specified by parameters a, b, c, d.

InterquartileRange[dist]

gives the difference between the upper and lower quartiles for the distribution dist.

# Details    • InterquartileRange is also known as IQR.
• InterquartileRange is a robust measure of dispersion, which means it is not very sensitive to outliers.
• InterquartileRange[data] is given by , where is given by Quartiles[data]. »
• • For MatrixQ data, the interquartile range is computed for each column vector with InterquartileRange[{{x1,y1,},{x2,y2,},}], equivalent to {InterquartileRange[{x1,x2,}],InterquartileRange[{y1,y2,}]}. »
• • For ArrayQ data, the interquartile range is equivalent to ArrayReduce[InterquartileRange,data,1]. »
• • InterquartileRange[data,{{a,b},{c,d}}] uses the Quartiles definition specified by parameters a, b, c, d. »
• Common choices of parameters {{a,b},{c,d}} include:
•  {{0, 0}, {1, 0}} inverse empirical CDF {{0, 0}, {0, 1}} linear interpolation (California method) {{1/2, 0}, {0, 0}} element numbered closest to p n {{1/2, 0}, {0, 1}} linear interpolation (hydrologist method; default) {{0, 1}, {0, 1}} mean‐based estimate (Weibull method) {{1, -1}, {0, 1}} mode‐based estimate {{1/3, 1/3}, {0, 1}} median‐based estimate {{3/8, 1/4}, {0, 1}} normal distribution estimate
• The default choice of parameters is {{1/2,0},{0,1}}. »
• The data can have the following additional forms and interpretations:
•  Association the values (the keys are ignored) » SparseArray as an array, equivalent to Normal[data] » QuantityArray quantities as an array » WeightedData based on the underlying EmpiricalDistribution » EventData based on the underlying SurvivalDistribution » TimeSeries, TemporalData, … vector or array of values (the time stamps ignored) » Image,Image3D RGB channel's values or grayscale intensity value » Audio amplitude values of all channels »
• InterquartileRange[dist] is given by , where is given by Quartiles[dist]. »
• • For a random process proc, the interquartile range function can be computed for a slice distribution at time t, SliceDistribution[proc,t], as InterquartileRange[SliceDistribution[proc,t]]. »
• # Examples

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## Basic Examples(2)

Interquartile range for a list of exact numbers:

Interquartile range of a parametric distribution:

## Scope(18)

### Basic Uses(8)

Exact input yields exact output:

Approximate input yields approximate output:

Compute results using other parametrizations:

Find the interquartile range for WeightedData:

Find the interquartile range for EventData:

Find the interquartile range for TemporalData:

Find the interquartile range of TimeSeries:

The interquartile range depends only on the values:

Find the interquartile range for data involving quantities:

### Array Data(5)

InterquartileRange for a matrix gives columnwise ranges:

Interquartile range for a tensor works across the first index:

Works with large arrays:

When the input is an Association, InterquartileRange works on its values:

SparseArray data can be used just like dense arrays:

Find interquartile range of a QuantityArray:

### Image and Audio Data(2)

Channelwise interquartile range values of an RGB image:

Interquartile range intensity value of a grayscale image:

Interquartile range amplitude of all channels:

### Distributions and Processes(3)

Find the interquartile range for a parametric distribution:

Interquartile range for a derived distribution:

Data distribution:

Interquartile range for a time slice of a random process:

## Applications(6)

InterquartileRange indicates the spread of values:

InterquartileRange can be used as a check for agreement between data and a distribution:

Generate a random sample:

Find the interquartile range of the data:

Compare with the interquartile range of the distribution:

Identify periods of high volatility in stock data using an annual moving interquartile range:

Find the interquartile ranges for the girth, height, and volume of timber, respectively, in 31 felled black cherry trees:

Compute InterquartileRange for slices of a collection of paths of a random process:

Choose a few slice times:

Plot of the interquartile range for the selected times:

Find the interquartile range of the heights for the children in a class:

Plot the interquartile range respective of the median:

## Properties & Relations(4)

InterquartileRange is the difference of linearly interpolated Quantile values:

InterquartileRange is the difference between the first and third quartiles:

QuartileDeviation is half the interquartile range:

BoxWhiskerChart shows the interquartile range for data:

## Possible Issues(1)

InterquartileRange requires numeric values in data: The symbolic closed form may exist for some distributions:

## Neat Examples(1)

The distribution of InterquartileRange estimates for 20, 100, and 300 samples: