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Wolfram Language & System Documentation Center
InterquartileRange
  • See Also
    • Quartiles
    • QuartileDeviation
    • MeanDeviation
    • StandardDeviation
    • Variance
    • TrimmedVariance
    • WinsorizedVariance
    • BiweightMidvariance
    • QnDispersion
    • SnDispersion
    • QuartileSkewness
    • Quantile
    • BoxWhiskerChart
  • Related Guides
    • Descriptive Statistics
    • Date & Time
    • Survival Analysis
    • Robust Descriptive Statistics
  • Tech Notes
    • Descriptive Statistics
    • See Also
      • Quartiles
      • QuartileDeviation
      • MeanDeviation
      • StandardDeviation
      • Variance
      • TrimmedVariance
      • WinsorizedVariance
      • BiweightMidvariance
      • QnDispersion
      • SnDispersion
      • QuartileSkewness
      • Quantile
      • BoxWhiskerChart
    • Related Guides
      • Descriptive Statistics
      • Date & Time
      • Survival Analysis
      • Robust Descriptive Statistics
    • Tech Notes
      • Descriptive Statistics

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
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Array Data  
Image and Audio Data  
Date and Time  
Distributions and Processes  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
History
Cite this Page
BUILT-IN SYMBOL
  • See Also
    • Quartiles
    • QuartileDeviation
    • MeanDeviation
    • StandardDeviation
    • Variance
    • TrimmedVariance
    • WinsorizedVariance
    • BiweightMidvariance
    • QnDispersion
    • SnDispersion
    • QuartileSkewness
    • Quantile
    • BoxWhiskerChart
  • Related Guides
    • Descriptive Statistics
    • Date & Time
    • Survival Analysis
    • Robust Descriptive Statistics
  • Tech Notes
    • Descriptive Statistics
    • See Also
      • Quartiles
      • QuartileDeviation
      • MeanDeviation
      • StandardDeviation
      • Variance
      • TrimmedVariance
      • WinsorizedVariance
      • BiweightMidvariance
      • QnDispersion
      • SnDispersion
      • QuartileSkewness
      • Quantile
      • BoxWhiskerChart
    • Related Guides
      • Descriptive Statistics
      • Date & Time
      • Survival Analysis
      • Robust Descriptive Statistics
    • Tech Notes
      • Descriptive Statistics

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:
  • Associationthe values (the keys are ignored) »
    SparseArrayas an array, equivalent to Normal[data] »
    QuantityArrayquantities as an array »
    WeightedDatabased on the underlying EmpiricalDistribution »
    EventDatabased on the underlying SurvivalDistribution »
    TimeSeries, TemporalData, …vector or array of values (the time stamps ignored) »
    Image,Image3DRGB channel's values or grayscale intensity value »
    Audioamplitude values of all channels »
    DateObject, TimeObjectlist of dates or list of times »
  • 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

open all close all

Basic Examples  (3)

Interquartile range for a list of exact numbers:

Interquartile range for a list of dates:

Interquartile range of a parametric distribution:

Scope  (22)

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:

Date and Time  (4)

Compute interquartile range of dates:

Compute the weighted interquartile range of dates:

Compare the simple interquartile range:

Compute the interquartile range of dates given in different calendars:

Compute the interquartile range of times:

List of times with different time zone specifications:

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:

See Also

Quartiles  QuartileDeviation  MeanDeviation  StandardDeviation  Variance  TrimmedVariance  WinsorizedVariance  BiweightMidvariance  QnDispersion  SnDispersion  QuartileSkewness  Quantile  BoxWhiskerChart

Tech Notes

    ▪
  • Descriptive Statistics

Related Guides

    ▪
  • Descriptive Statistics
  • ▪
  • Date & Time
  • ▪
  • Survival Analysis
  • ▪
  • Robust Descriptive Statistics

History

Introduced in 2007 (6.0) | Updated in 2017 (11.1) ▪ 2023 (13.3) ▪ 2024 (14.1)

Wolfram Research (2007), InterquartileRange, Wolfram Language function, https://reference.wolfram.com/language/ref/InterquartileRange.html (updated 2024).

Text

Wolfram Research (2007), InterquartileRange, Wolfram Language function, https://reference.wolfram.com/language/ref/InterquartileRange.html (updated 2024).

CMS

Wolfram Language. 2007. "InterquartileRange." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/InterquartileRange.html.

APA

Wolfram Language. (2007). InterquartileRange. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InterquartileRange.html

BibTeX

@misc{reference.wolfram_2025_interquartilerange, author="Wolfram Research", title="{InterquartileRange}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/InterquartileRange.html}", note=[Accessed: 01-December-2025]}

BibLaTeX

@online{reference.wolfram_2025_interquartilerange, organization={Wolfram Research}, title={InterquartileRange}, year={2024}, url={https://reference.wolfram.com/language/ref/InterquartileRange.html}, note=[Accessed: 01-December-2025]}

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