InterquartileRange

✖
`InterquartileRange`

Updated in 14.1

✖

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 »
	DateObject, TimeObject	list 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 allclose all

Basic Examples (3)Summary of the most common use cases

Interquartile range for a list of exact numbers:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cho

Out[1]=1

Interquartile range for a list of dates:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-t7ech

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ziof1v

Out[2]=2

Interquartile range of a parametric distribution:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b133z2

Out[1]=1

Scope (22)Survey of the scope of standard use cases

Basic Uses (8)

Exact input yields exact output:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ug7y2

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-bcry2t

Out[2]=2

Approximate input yields approximate output:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cg1nsz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-dvod55

Out[2]=2

Compute results using other parametrizations:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-dl6aeg

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b8keb3

Out[2]=2

Find the interquartile range for WeightedData:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-d0wc9z

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-f1vfw

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-qyv0h

Out[3]=3

Find the interquartile range for EventData:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-e67u14

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-or2nrz

Out[2]=2

Find the interquartile range for TemporalData:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-gx0rsr

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-f5ij1t

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-8e999

Out[3]=3

Find the interquartile range of TimeSeries:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-iqmjm1

Out[1]=1

The interquartile range depends only on the values:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-i4d98s

Out[2]=2

Find the interquartile range for data involving quantities:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-jopin9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-e8c21s

Out[2]=2

Array Data (5)

InterquartileRange for a matrix gives columnwise ranges:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-eymwwi

Out[1]=1

Interquartile range for a tensor works across the first index:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-lw96ov

Out[2]=2

Works with large arrays:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-nknun

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ma3v2m

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cs7n5q

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-xmyxp8

Out[2]=2

SparseArray data can be used just like dense arrays:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-f5ekjj

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-c4gal2

Out[2]=2

Find interquartile range of a QuantityArray:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-lgwnaj

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-k03qc6

Out[2]=2

Image and Audio Data (2)

Channelwise interquartile range values of an RGB image:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-hfby9q

Out[1]=1

Interquartile range intensity value of a grayscale image:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ue2gq5

Out[2]=2

Interquartile range amplitude of all channels:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-nq1jnz

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-bs38vd

Out[2]=2

Date and Time (4)

Compute interquartile range of dates:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b1smxx

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-pa4nmn

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-uok1il

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-baj2x3

Out[4]=4

Compute the weighted interquartile range of dates:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-c98kbd

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-8c1had

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-t71b2h

Out[3]=3

Compare the simple interquartile range:

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-w3hn8w

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-3rxtf0

Out[5]=5

Compute the interquartile range of dates given in different calendars:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-wbzcuv

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-9ius88

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-qe5gbw

Out[3]=3

Compute the interquartile range of times:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-et9bla

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ztsexm

Out[2]=2

List of times with different time zone specifications:

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-mrqghz

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-1d7sk5

Out[4]=4

Distributions and Processes (3)

Find the interquartile range for a parametric distribution:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-hbq28j

Out[1]=1

Interquartile range for a derived distribution:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-rgc72x

Out[1]=1

Data distribution:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-215ry

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-fq5ptk

Out[3]=3

Interquartile range for a time slice of a random process:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-c4ojmv

Out[1]=1

Applications (6)Sample problems that can be solved with this function

InterquartileRange indicates the spread of values:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cmvg05

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b7qo6

Out[2]=2

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-6i89hv

Generate a random sample:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-yyz1hn

Find the interquartile range of the data:

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-pfpq4z

Out[3]=3

Compare with the interquartile range of the distribution:

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-6v20t1

Out[4]=4

In[5]:=5

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-upwvgd

Out[5]=5

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-0tl2ek

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-kfgcti

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-bef0x

Out[3]=3

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-iy7bhd

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-o0jkg

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-mo39rs

Out[3]=3

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cgixp7

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-8se1zg

Choose a few slice times:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-52xxug

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-iakfqb

Plot of the interquartile range for the selected times:

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-5u9dct

Out[4]=4

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cevfij

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-fllmtw

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-celepo

Out[3]=3

Plot the interquartile range respective of the median:

In[4]:=4

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-g98mgx

Out[4]=4

Properties & Relations (4)Properties of the function, and connections to other functions

InterquartileRange is the difference of linearly interpolated Quantile values:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-chohu9

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-cuwfxn

Out[3]=3

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-kqldeg

Out[2]=2

InterquartileRange is the difference between the first and third quartiles:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-d8ys7g

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-ikpxf8

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-fs2ypr

Out[3]=3

QuartileDeviation is half the interquartile range:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b8uymf

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-h8t71a

Out[2]=2

In[3]:=3

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-n0cv2

Out[3]=3

BoxWhiskerChart shows the interquartile range for data:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-i5d4c7

Out[1]=1

Possible Issues (1)Common pitfalls and unexpected behavior

InterquartileRange requires numeric values in data:

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-g88

Out[1]=1

The symbolic closed form may exist for some distributions:

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-hsjsiw

Out[2]=2

Neat Examples (1)Surprising or curious use cases

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

In[1]:=1

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-b3v7m9

Out[1]=1

In[2]:=2

✖

https://wolfram.com/xid/0jvlvbinvb11tvdiso0a-8raem

Out[2]=2

Top

More Learning

Tech Support

Wolfram Solutions

Wolfram Solutions For Education

Get Started

Grow Your Skills

Work with Us

Educational Programs for Adults

Educational Programs for Youth

Read

InterquartileRange

✖
`InterquartileRange`

Details

Examples

Basic Examples (3)Summary of the most common use cases

Scope (22)Survey of the scope of standard use cases

Basic Uses (8)

Array Data (5)

Image and Audio Data (2)

Date and Time (4)

Distributions and Processes (3)

Applications (6)Sample problems that can be solved with this function

Properties & Relations (4)Properties of the function, and connections to other functions

Possible Issues (1)Common pitfalls and unexpected behavior

Neat Examples (1)Surprising or curious use cases

Text

CMS

APA

BibTeX

BibLaTeX

InterquartileRange ✖ InterquartileRange

Details

Examples

Basic Examples (3)Summary of the most common use cases

Scope (22)Survey of the scope of standard use cases

Basic Uses (8)

Array Data (5)

Image and Audio Data (2)

Date and Time (4)

Distributions and Processes (3)

Applications (6)Sample problems that can be solved with this function

Properties & Relations (4)Properties of the function, and connections to other functions

Possible Issues (1)Common pitfalls and unexpected behavior

Neat Examples (1)Surprising or curious use cases

See Also

Tech Notes

Related Guides

History

Text

CMS

APA

BibTeX

BibLaTeX

InterquartileRange

✖
`InterquartileRange`