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HistogramList
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HistogramList

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HistogramList[{x1,x2,…}]

gives a list of bins and histogram heights of the values xi.

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HistogramList[{{x1,y1,…},{x2,y2,…},…}]

gives a list of bins and histogram heights of the values {xi,yi,…}.

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HistogramList[…,bspec]

gives a list of bins and histogram heights with bins specified by bspec.

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HistogramList[…,bspec,hspec]

gives a list of bins and histogram heights with bin heights computed according to the specification hspec.

Details and Options

  • HistogramList is also known as quadrat counts.
  • HistogramList produces a list of bin delimiters {b1,b2,…} for each dimension and a depth- array of values for each bin.
  • For 2D data, the output has the form {{{bx1,bx2,…},{by1,…}},{{v1,1,v1,2,…},{v2,1,…},…}} where vi,j is the value corresponding to the bin [bxi,bxi+1)×[byj,byj+1).
  • HistogramList[data] by default gives equal bins chosen to approximate an assumed underlying smooth distribution of the values {xi,yi,…}.
  • The width of each bin is computed according to the values xi, the width according to the yi, etc.
  • The following bin specifications bpsec can be given:
  • nuse n bins
    {w}use bins of width w
    {min,max,w}use bins of width w from min to max
    {{b1,b2,…}}use bins [b1,b2),[b2,b3),…
    Automaticdetermine bin widths automatically
    "name"use a named binning method
    {"Log",bspec}apply binning bspec on log-transformed data
    fwapply fw to get an explicit bin specification {b1,b2,…}
    {xspec,yspec,…}give different x, y, etc. specifications
  • The binning specification "Log" is taken to use the Automatic underlying binning method.
  • Possible named binning methods include:
  • "Sturges"compute the number of bins based on the length of data
    "Scott"asymptotically minimize the mean square error
    "FreedmanDiaconis"twice the interquartile range divided by the cube root of sample size
    "Knuth"balance likelihood and prior probability of a piecewise uniform model
    "Wand"one-level recursive approximate Wand binning
  • The function fb in HistogramList[data,fb] is applied to a list of all {xi,yi,…}, and should return an explicit bin list {{bx1,bx2,…},{by1,by2,…},…}. In HistogramList[data,{fx,fy,…}], fx is applied to the list of xi, and fy to the list of yi, etc.
  • Different forms of histogram data can be obtained by giving different bin height specifications hspec in HistogramList[data,bspec,hspec]. The following forms can be used:
  • "Count"the number of values lying in each bin
    "CumulativeCount"cumulative counts
    "SurvivalCount"survival counts
    "Probability"fraction of values lying in each bin
    "PDF"probability density function
    "CDF"cumulative distribution function
    "SF"survival function
    "HF"hazard function
    "CHF"cumulative hazard function
    fhheights obtained by applying fh to bins and counts
  • Only values {xi,yi,…} that consist of real numbers are assigned to bins; others are taken to be missing.
  • The function fh in HistogramList[data,bspec,fh] is applied to two arguments: a list of bin delimiters {{bx1,bx2,…},{by1,by2,…},…} and for -dimensional data an array of depth with counts for each bin.

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Generate a list of bin delimiters and counts for a dataset:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-vhwcy8
Out[1]=1

Generate a list of bin delimiters and counts for a bivariate dataset:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-rvz604
Out[1]=1

Use unit width bins:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-g8bszr
Out[1]=1

Use different height functions:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-tyzzu0
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-6r9vba
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-38iqvl
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-vmxffw
Out[4]=4

Scope  (10)Survey of the scope of standard use cases

Bin Specifications  (7)

Specify the number of bins to use:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-bnll24
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-fu5ydu
Out[2]=2

Specify the bin width:

In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-jhaknu
Out[3]=3

Specify limits and bin size:

In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-cseaq8
Out[4]=4

List specific bin delimiters:

In[5]:=5
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https://wolfram.com/xid/0dc26n1gvnwa-bczne
Out[5]=5

Bins for discrete values are centered on the values when possible:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-fwau0w
Out[1]=1

Specify the number of bins to use for each dimension:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-xzyje7
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-2fz6vi
Out[2]=2

Specify different bin specifications for each dimension:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-ig2hij
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-meo6go
Out[2]=2

Use different named binning methods:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-dqt52g
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-m72bmm
Out[2]=2

Use logarithmically spaced bins:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-oews3c
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-gnf3tc
Out[2]=2

Delimit bins on integer boundaries using a binning function:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-e8jw76
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-gkihwi
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-bzo34f
Out[3]=3

Height Specifications  (3)

Use different height specifications:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-cphhlw
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-2aqgi8
Out[2]=2

Use a height function that accumulates the bin counts:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-o40mc5
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-ki7l92
In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-jaap33
Out[3]=3

Nonreal data is taken to be missing:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-bibvmw
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-4jj9vl
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

Functions such as Histogram and Histogram3D use HistogramList to compute bins and heights:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-t3lyka
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-fvzvxu
Out[2]=2

BinCounts and BinLists can be used to find items in specific bins:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-5d8nnf
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-4xgmnu
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-ursv17
Out[3]=3

HistogramList also provides the actual bins:

In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-kzcczf
Out[4]=4

Bivariate data is given as a list of pairs of numbers:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-s7e1rf
Out[1]=1

There is one set of bins for each variable dimension:

In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-bb175j
Out[2]=2

The array depth of the counts is the same as the number of variable dimensions:

In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-s42tky
Out[3]=3

With data consisting of 5-tuples:

In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-w0xdfm
In[5]:=5
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https://wolfram.com/xid/0dc26n1gvnwa-l52900
Out[5]=5

For bin delimiters, there are bins in each dimension:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-n12g2
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-mksmny
Out[2]=2

HistogramList ignores the times if the input is TimeSeries:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-8pf3w2
In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-fh030f
Out[2]=2

Compare to the histogram distribution of the values:

In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-vy8bf1
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-60bp7w
Out[4]=4

Possible Issues  (4)Common pitfalls and unexpected behavior

Bins include the left endpoint, but not the right, which can result in unexpected bins:

In[2]:=2
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https://wolfram.com/xid/0dc26n1gvnwa-4uturq
In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-fj26av
Out[3]=3

The value 1 is not included in this histogram because it would be in the bin [1,1.2):

In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-mm1op8
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0dc26n1gvnwa-x8gyp7
Out[5]=5

The bins obtained using the specification {min,max,w} may not include all data points:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-ijov4e
In[2]:=2
✖
https://wolfram.com/xid/0dc26n1gvnwa-swfvbx
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0dc26n1gvnwa-n7b6us
Out[3]=3

Specify the min and max so all data points are included in the bin covering:

In[4]:=4
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https://wolfram.com/xid/0dc26n1gvnwa-lv5ut2
In[5]:=5
✖
https://wolfram.com/xid/0dc26n1gvnwa-42jqa4
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0dc26n1gvnwa-xcrjii
Out[6]=6

Alternatively, use the width-only bin specification {w}:

In[7]:=7
✖
https://wolfram.com/xid/0dc26n1gvnwa-45gh2w
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0dc26n1gvnwa-qbvho6
Out[8]=8

Results for multidimensional data can get quite large:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-vkyh3e
Out[1]=1

Most of the values are zero:

In[2]:=2
✖
https://wolfram.com/xid/0dc26n1gvnwa-r9h3gt
Out[2]=2

HistogramList does not recognize the path structure of multipath TemporalData input:

In[1]:=1
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https://wolfram.com/xid/0dc26n1gvnwa-v4xauw
In[2]:=2
✖
https://wolfram.com/xid/0dc26n1gvnwa-fssoy6
Out[2]=2

Find HistogramList pathwise:

In[3]:=3
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https://wolfram.com/xid/0dc26n1gvnwa-v83h2w
Out[3]=3
Wolfram Research (2010), HistogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/HistogramList.html.
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Wolfram Research (2010), HistogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/HistogramList.html.

Text

Wolfram Research (2010), HistogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/HistogramList.html.

✖
Wolfram Research (2010), HistogramList, Wolfram Language function, https://reference.wolfram.com/language/ref/HistogramList.html.

CMS

Wolfram Language. 2010. "HistogramList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HistogramList.html.

✖
Wolfram Language. 2010. "HistogramList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HistogramList.html.

APA

Wolfram Language. (2010). HistogramList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HistogramList.html

✖
Wolfram Language. (2010). HistogramList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HistogramList.html

BibTeX

@misc{reference.wolfram_2025_histogramlist, author="Wolfram Research", title="{HistogramList}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/HistogramList.html}", note=[Accessed: 26-March-2025 ]}

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@misc{reference.wolfram_2025_histogramlist, author="Wolfram Research", title="{HistogramList}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/HistogramList.html}", note=[Accessed: 26-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_histogramlist, organization={Wolfram Research}, title={HistogramList}, year={2010}, url={https://reference.wolfram.com/language/ref/HistogramList.html}, note=[Accessed: 26-March-2025 ]}

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@online{reference.wolfram_2025_histogramlist, organization={Wolfram Research}, title={HistogramList}, year={2010}, url={https://reference.wolfram.com/language/ref/HistogramList.html}, note=[Accessed: 26-March-2025 ]}