Editing Multimodal distribution

{{Short description|Probability distribution with more than one mode}}
{{redirect|Bimodal|the musical concept|Bimodality}}

[[Image:Bimodal.png|thumb|'''Figure 1.''' A simple bimodal distribution, in this case a [[mixture distribution|mixture]] of two [[normal distribution]]s with the same variance but different means.  The figure shows the [[probability density function]] (p.d.f.), which is an equally-weighted average of the bell-shaped p.d.f.s of the two normal distributions. If the weights were not equal, the resulting distribution could still be bimodal but with peaks of different heights.]]
[[File:Bimodal geological.PNG|thumb|'''Figure 2.''' A bimodal distribution.]]
[[Image:Bimodal-bivariate-small.png|thumb|'''Figure 3.''' A bivariate, multimodal distribution]]
[[File:Unimodal Nonmonotonic Distribution.png|thumb|alt=A 3D plot of a probability distribution. It ripples and spirals away from the origin, with only one local maximum near the origin.|'''Figure 4.''' A non-example: a ''[[unimodal]]'' distribution, that would become multimodal if conditioned on either x or y.]]

In [[statistics]], a '''multimodal''' '''distribution''' is a [[probability distribution]] with more than one [[mode (statistics)|mode]] (i.e., more than one local peak of the distribution). These appear as distinct peaks (local maxima) in the [[probability density function]], as shown in Figures 1 and 2. Categorical, continuous, and discrete data can all form multimodal distributions. Among univariate analyses, multimodal distributions are commonly bimodal.{{Citation needed|date=July 2022}}

==Terminology==

When the two modes are unequal the larger mode is known as the major mode and the other as the minor mode. The least frequent value between the modes is known as the [[antimode]]. The difference between the major and minor modes is known as the [[amplitude]]. In time series the major mode is called the [[acrophase]] and the antimode the [[batiphase]].{{citation needed|date=May 2017}}

==Galtung's classification==

Galtung introduced a classification system (AJUS) for distributions:<ref name=Galtung1969>{{cite book |last=Galtung |first=J. |year=1969 |title=Theory and methods of social research |publisher=Universitetsforlaget |location=Oslo |isbn=0-04-300017-7 }}</ref>

*A: unimodal distribution – peak in the middle
*J: unimodal – peak at either end 
*U: bimodal – peaks at both ends
*S: bimodal or multimodal – multiple peaks

This classification has since been modified slightly:

*J: (modified) – peak on right
*L: unimodal – peak on left 
*F: no peak (flat)

Under this classification bimodal distributions are classified as type S or U.

==Examples==

Bimodal distributions occur both in mathematics and in the natural sciences.

===Probability distributions===

Important bimodal distributions include the [[arcsine distribution]] and the [[beta distribution]] (iff both parameters ''a'' and ''b'' are less than 1). Others include the [[U-quadratic distribution]].

The ratio of two normal distributions is also bimodally distributed. Let

<math display="block"> R = \frac{ a + x }{ b + y } </math>

where ''a'' and ''b'' are constant and ''x'' and ''y'' are distributed as normal variables with a mean of 0 and a standard deviation of 1. ''R'' has a known density that can be expressed as a [[confluent hypergeometric function]].<ref name=Fieller1932>{{cite journal |author=Fieller E |date=1932 |title=The distribution of the index in a normal bivariate population |journal=Biometrika |volume=24 |issue=3–4 |pages=428–440 |doi=10.1093/biomet/24.3-4.428}}</ref>

The distribution of the [[Inverse distribution|reciprocal]] of a ''t'' distributed random variable is bimodal when the degrees of freedom are more than one. Similarly the reciprocal of a normally distributed variable is also bimodally distributed.

A ''t'' statistic generated from data set drawn from a [[Cauchy distribution]] is bimodal.<ref name=Fiorio2010>{{cite journal | last1 = Fiorio | first1 = CV | last2 = HajivassILiou | first2 = VA | last3 = Phillips | first3 = PCB | year = 2010 | title = Bimodal t-ratios: the impact of thick tails on inference | url = https://ink.library.smu.edu.sg/soe_research/1817| journal = The Econometrics Journal | volume = 13 | issue = 2| pages = 271–289 | doi = 10.1111/j.1368-423X.2010.00315.x | s2cid = 363740 }}</ref>

===Occurrences in nature===

Examples of variables with bimodal distributions include the time between eruptions of certain [[geyser]]s, the [[Galaxy color-magnitude diagram|color of galaxies]], the size of worker [[weaver ants]], the age of incidence of [[Hodgkin's lymphoma]], the speed of inactivation of the drug [[isoniazid]] in US adults, the absolute magnitude of [[nova]]e, and the [[Circadian rhythm|circadian activity patterns]] of those [[crepuscular]] animals that are active both in morning and evening twilight. In fishery science multimodal length distributions reflect the different year classes and can thus be used for age distribution- and growth estimates of the fish population.<ref>[http://www.fao.org/docrep/W5449E/w5449e05.htm.|FAO: Introduction to tropical fish stock assessment]</ref> Sediments are usually distributed in a bimodal fashion. When sampling mining galleries crossing either the host rock and the mineralized veins, the distribution of geochemical variables would be bimodal. Bimodal distributions are also seen in traffic analysis, where traffic peaks in during the AM rush hour and then again in the PM rush hour. This phenomenon is also seen in daily water distribution, as water demand, in the form of showers, cooking, and toilet use, generally peak in the morning and evening periods. Some genes in bacteria have also exhibited bimodal distributions of gene expression both in normal as well as in stress conditions.<ref name=iscb>{{Cite journal |last=Baptista |first=Ines S. C. |last2=Dash |first2=Suchintak |last3=Arsh |first3=Amir M. |last4=Kandavalli |first4=Vinodh |last5=Scandolo |first5=Carlo Maria |last6=Sanders |first6=Barry C. |last7=Ribeiro |first7=Andre S. |date=2025-02-13 |title=Bimodality in E. coli gene expression: Sources and robustness to genome-wide stresses |url=https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1012817 |journal=PLOS Computational Biology |language=en |volume=21 |issue=2 |pages=e1012817 |doi=10.1371/journal.pcbi.1012817 |issn=1553-7358 |pmc=11825099 |doi-access=free}}</ref>

===Econometrics===

In [[econometrics|econometric]] models, the parameters may be bimodally distributed.<ref name=Phillips2006>{{cite journal |last=Phillips |first=P. C. B. |author-link=Peter C. B. Phillips |year=2006 |title=A remark on bimodality and weak instrumentation in structural equation estimation |journal=[[Econometric Theory]] |volume=22 |issue=5 |pages=947–960 |doi=10.1017/S0266466606060439 |s2cid=16775883 |url=http://cowles.yale.edu/sites/default/files/files/pub/d15/d1540.pdf }}</ref>

==Origins==

{{main article|Mixture distribution}}

===Mathematical===

A bimodal distribution commonly arises as a mixture of two different [[unimodal]] distributions (i.e. distributions having only one mode).  In other words, the bimodally distributed random variable X is defined as <math> Y </math> with probability <math> \alpha </math> or <math> Z </math> with probability <math> (1-\alpha), </math> where ''Y'' and ''Z'' are unimodal random variables and <math>0 < \alpha < 1</math> is a mixture coefficient.

Mixtures with two distinct components need not be bimodal and two component mixtures of unimodal component densities can have more than two modes. There is no immediate connection between the number of components in a mixture and the number of modes of the resulting density.

===Particular distributions===

Bimodal distributions, despite their frequent occurrence in data sets, have only rarely been studied{{citation needed|date=March 2019}}. This may be because of the difficulties in estimating their parameters either with frequentist or Bayesian methods. Among those that have been studied are

* Bimodal exponential distribution.<ref name=Hassan2010>{{cite journal | last1 = Hassan | first1 = MY | last2 = Hijazi | first2 = RH | year = 2010 | title = A bimodal exponential power distribution | journal = Pakistan Journal of Statistics | volume = 26 | issue = 2| pages = 379–396 }}</ref>
* Alpha-skew-normal distribution.<ref name=Elal-Olivero2010>{{cite journal | last1 = Elal-Olivero | first1 = D | year = 2010 | title = Alpha-skew-normal distribution | journal = Proyecciones Journal of Mathematics | volume = 29 | issue = 3| pages = 224–240 | doi=10.4067/s0716-09172010000300006| doi-access = free }}</ref>
* Bimodal skew-symmetric normal distribution.<ref name=Hassan2013>{{cite journal |last1=Hassan |first1=M. Y. |last2=El-Bassiouni |first2=M. Y. |year=2016 |title=Bimodal skew-symmetric normal distribution |journal=Communications in Statistics - Theory and Methods |volume=45 |issue=5 |pages=1527–1541 |doi=10.1080/03610926.2014.882950 |s2cid=124087015 }}</ref>
* A mixture of [[Conway-Maxwell-Poisson distribution]]s has been fitted to bimodal count data.<ref name=Bosea2013>{{cite book |last1=Bosea |first1=S. |last2=Shmuelib |first2=G. |last3=Sura |first3=P. |last4=Dubey |first4=P. |year=2013 |chapter=Fitting Com-Poisson mixtures to bimodal count data |title=Proceedings of the 2013 International Conference on Information, Operations Management and Statistics (ICIOMS2013), Kuala Lumpur, Malaysia |pages=1–8 |chapter-url=https://www.galitshmueli.com/system/files/ICIOMS%202013%20Malaysia%20Paper%20ID%2028.pdf }}</ref>

Bimodality also naturally arises in the [[Catastrophe theory#Cusp catastrophe|cusp catastrophe distribution]].

===Biology===

In biology, several factors are known to contribute to bimodal distributions of population sizes{{citation needed|date=March 2019}}:

*the initial distribution of individual sizes
*the distribution of growth rates among the individuals
*the size and time dependence of the growth rate of each individual
* mortality rates that may affect each size class differently
* the DNA methylation in human and mouse genome.
* the dynamics of transcription at the promoter region.

The bimodal distribution of sizes of [[weaver ant]] workers arises due to existence of two distinct classes of workers, namely major workers and minor workers.<ref name="Weber1946">{{cite journal|author=Weber, NA|year=1946| title=Dimorphism in the African ''Oecophylla'' worker and an anomaly (Hym.: Formicidae)| journal=Annals of the Entomological Society of America| volume=39| pages=7–10| url=http://antbase.org/ants/publications/10434/10434.pdf| doi=10.1093/aesa/39.1.7}}</ref>

The [[distribution of fitness effects]] of mutations for both whole [[genome]]s<ref>{{cite journal|last=Sanjuán|first=R|title=Mutational fitness effects in RNA and single-stranded DNA viruses: common patterns revealed by site-directed mutagenesis studies.|journal=Philosophical Transactions of the Royal Society of London B: Biological Sciences|date=Jun 27, 2010|volume=365|issue=1548|pages=1975–82|pmid=20478892|doi=10.1098/rstb.2010.0063|pmc=2880115}}</ref><ref>{{cite journal|last=Eyre-Walker|first=A|author2=Keightley, PD|title=The distribution of fitness effects of new mutations.|journal=Nature Reviews Genetics|date=Aug 2007|volume=8|issue=8|pages=610–8|pmid=17637733|doi=10.1038/nrg2146|s2cid=10868777}}</ref> and individual [[gene]]s<ref>{{cite journal|last=Hietpas|first=RT|author2=Jensen, JD |author3=Bolon, DN |title=Experimental illumination of a fitness landscape.|journal=Proceedings of the National Academy of Sciences of the United States of America|date=May 10, 2011|volume=108|issue=19|pages=7896–901|pmid=21464309|doi=10.1073/pnas.1016024108|pmc=3093508|bibcode = 2011PNAS..108.7896H |doi-access=free}}</ref> is also frequently found to be bimodal with most [[mutations]] being either neutral or lethal with relatively few having intermediate effect.

==General properties==

A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their [[standard deviation]]s to produce bimodality when the two distribution curves are combined.<ref name="Schilling2002">{{Cite journal|title=Is Human Height Bimodal?|first1=Mark F. |last1=Schilling |first2= Ann E.| last2=Watkins | author2-link = Ann E. Watkins |first3=William |last3=Watkins| journal=[[The American Statistician]]| doi=10.1198/00031300265 |volume=56 |year=2002| pages=223–229 |issue=3|s2cid=53495657 }}</ref>

Bimodal distributions have the peculiar property that – unlike the unimodal distributions – the mean may be a more robust sample estimator than the median.<ref name=Mosteller1977>{{cite book |last1=Mosteller |first1=F. |last2=Tukey |first2=J. W. |year=1977 |title=Data Analysis and Regression: A Second Course in Statistics |location=Reading, Mass |publisher=Addison-Wesley |isbn=0-201-04854-X }}</ref> This is clearly the case when the distribution is U-shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails.

===Moments of mixtures===

Let

<math display="block"> f( x ) = p g_1( x ) + ( 1 - p ) g_2( x ) \, </math>

where {{math|''g''<sub>''i''</sub>}} is a probability distribution and {{math|''p''}} is the mixing parameter.

The moments of {{math|''f''(''x'')}} are<ref name=Kim2003>{{cite web |last1=Kim |first1=T.-H. |last2=White |first2=H. |author-link2=Halbert White |year=2003 |url=https://escholarship.org/uc/item/7b52v07p |title=On more robust estimation of skewness and kurtosis: Simulation and application to the S & P 500 index }}</ref>

<math display="block">\begin{align}
\mu &= p \mu_1 + ( 1 - p ) \mu_2 \\[1ex]
\nu_2 &= p \left[ \sigma_1^2 + \delta_1^2 \right] + ( 1 - p ) \left[ \sigma_2^2 + \delta_2^2 \right] \\[1ex]
\nu_3 &= p \left[ S_1 \sigma_1^3 + 3 \delta_1 \sigma_1^2 + \delta_1^3 \right] + ( 1 - p ) \left[ S_2 \sigma_2^3 + 3 \delta_2 \sigma_2^2 + \delta_2^3 \right] \\[1ex]
\nu_4 &= p \left[ K_1 \sigma_1^4 + 4 S_1 \delta_1 \sigma_1^3 + 6 \delta_1^2 \sigma_1^2 + \delta_1^4 \right] + ( 1 - p ) \left[ K_2 \sigma_2^4 + 4 S_2 \delta_2 \sigma_2^3 + 6 \delta_2^2 \sigma_2^2 + \delta_2^4 \right] \\
\end{align}</math>

where 
* <math> \mu = \int x f( x ) \, dx </math>
* <math> \delta_i = \mu_i - \mu </math>
* <math> \nu_r = \int ( x - \mu )^r f( x ) \, dx </math>

and {{math|''S''<sub>''i''</sub>}} and {{math|''K''<sub>''i''</sub>}} are the [[skewness]] and [[kurtosis]] of the {{mvar|i}}-th distribution.

==Mixture of two normal distributions==

It is not uncommon to encounter situations where an investigator believes that the data comes from a mixture of two normal distributions. Because of this, this mixture has been studied in some detail.<ref name=Robertson1969>{{cite journal | last1 = Robertson | first1 = CA | last2 = Fryer | first2 = JG | year = 1969 | title = Some descriptive properties of normal mixtures | journal = Skandinavisk Aktuarietidskrift | volume = 69 | issue = 3–4| pages = 137–146 |doi=10.1080/03461238.1969.10404590}}</ref>

A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two [[normal distribution]]s with equal [[standard deviation]]s is bimodal only if their means differ by at least twice the common standard deviation.<ref name="Schilling2002"/> Estimates of the parameters is simplified if the variances can be assumed to be equal (the [[homoscedastic]] case).

If the means of the two normal distributions are equal, then the combined distribution is unimodal. Conditions for [[unimodality]] of the combined distribution were derived by Eisenberger.<ref name=Eisenberger1964>{{cite journal | last1 = Eisenberger | first1 = I | year = 1964 | title = Genesis of bimodal distributions | journal = Technometrics | volume = 6 | issue = 4| pages = 357–363 | doi=10.1080/00401706.1964.10490199}}</ref> Necessary and sufficient conditions for a mixture of normal distributions to be bimodal have been identified by Ray and Lindsay.<ref name=Ray2005>{{cite journal | last1 = Ray | first1 = S | last2 = Lindsay | first2 = BG | year = 2005 | title = The topography of multivariate normal mixtures | journal = Annals of Statistics | volume = 33 | issue = 5| pages = 2042–2065 | doi=10.1214/009053605000000417| arxiv = math/0602238 | s2cid = 36234163 }}</ref>

A mixture of two approximately equal mass normal distributions has a negative kurtosis since the two modes on either side of the center of mass effectively reduces the tails of the distribution.

A mixture of two normal distributions with highly unequal mass has a positive kurtosis since the smaller distribution lengthens the tail of the more dominant normal distribution.

Mixtures of other distributions require additional parameters to be estimated.

===Tests for unimodality===

*When the components of the mixture have equal variances the mixture is unimodal [[if and only if]]<ref name=Holzmann2008>{{cite journal | last1 = Holzmann | first1 = Hajo | last2 = Vollmer | first2 = Sebastian | year = 2008 | title = A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU | journal = AStA Advances in Statistical Analysis | volume = 2 | issue = 1| pages = 57–69 | doi=10.1007/s10182-008-0057-2 | s2cid = 14470055 | url = http://resolver.sub.uni-goettingen.de/purl?gs-1/8526 }}</ref> <math display="block"> d \le 1 </math> or <math display="block">\left\vert \log( 1 - p ) - \log( p ) \right\vert \ge 2 \log( d - \sqrt{ d^2 - 1 } ) + 2d \sqrt{ d^2 - 1 } ,</math> where ''p'' is the mixing parameter and <math display="block"> d = \frac{ \left\vert \mu_1 - \mu_2 \right\vert }{ 2 \sigma }, </math> and where ''μ''<sub>1</sub> and ''μ''<sub>2</sub> are the means of the two normal distributions and ''σ'' is their standard deviation.
*The following test for the case ''p'' = 1/2 was described by Schilling ''et al''.<ref name=Schilling2002/> Let <math display="block"> r = \frac{ \sigma_1^2 }{ \sigma_2^2 } .</math> The separation factor (''S'') is <math display="block"> S = \frac{ \sqrt{ -2 + 3r + 3r^2 - 2r^3 + 2 \left( 1 - r + r^2 \right)^{ 1.5 } } }{ \sqrt{ r } \left( 1 + \sqrt{ r } \right) } .</math> If the variances are equal then ''S'' = 1. The mixture density is unimodal if and only if <math display="block"> | \mu_1 - \mu_2 | < S | \sigma_1 + \sigma_2 | .</math>
*A sufficient condition for unimodality is<ref name=Behboodian1970>{{cite journal | last1 = Behboodian | first1 = J | year = 1970 | title = On the modes of a mixture of two normal distributions | journal = Technometrics | volume = 12 | issue = 1| pages = 131–139 | doi=10.2307/1267357| jstor = 1267357 }}</ref><math display="block">|\mu_1-\mu_2| \le2\min (\sigma_1,\sigma_2).</math>
*If the two normal distributions have equal standard deviations <math>\sigma,</math> a sufficient condition for unimodality is<ref name=Behboodian1970/><math display="block">|\mu _1-\mu_2|\le 2\sigma \sqrt{1+\frac{ \left|\ln p-\ln (1-p)\right|}{2}}.</math>

==Summary statistics==

Bimodal distributions are a commonly used example of how summary statistics such as the [[mean]], [[median]], and [[standard deviation]] can be deceptive when used on an arbitrary distribution. For example, in the distribution in Figure 1, the mean and median would be about zero, even though zero is not a typical value.  The standard deviation is also larger than deviation of each normal distribution.

Although several have been suggested, there is no presently generally agreed summary statistic (or set of statistics) to quantify the parameters of a general bimodal distribution. For a mixture of two normal distributions the means and standard deviations along with the mixing parameter (the weight for the combination) are usually used – a total of five parameters.

===Ashman's D===

A statistic that may be useful is Ashman's D:<ref name=Ashman1994>{{cite journal |author1=Ashman KM |author2=Bird CM |author3=Zepf SE |date=1994 |title=Detecting bimodality in astronomical datasets |journal=The Astronomical Journal |volume=108 |pages=2348–2361 |arxiv=astro-ph/9408030 |doi=10.1086/117248 |bibcode=1994AJ....108.2348A|s2cid=13464256 }}</ref>

<math display="block"> D = \frac{ \left| \mu_1 - \mu_2 \right| }{ \sqrt{ 2 \left( \sigma_1^2 + \sigma_2^2 \right) } } </math>

where ''μ''<sub>1</sub>, ''μ''<sub>2</sub> are the means and ''σ''<sub>1</sub>, ''σ''<sub>2</sub> are the standard deviations.

For a mixture of two normal distributions ''D'' > 2 is required for a clean separation of the distributions.

===van der Eijk's A===

This measure is a weighted average of the degree of agreement the frequency distribution.<ref name=Van_der_Eijk2001>{{cite journal | last1 = Van der Eijk | first1 = C | year = 2001 | title = Measuring agreement in ordered rating scales | journal = Quality & Quantity | volume = 35 | issue = 3| pages = 325–341 | doi=10.1023/a:1010374114305| s2cid = 189822180 }}</ref> ''A'' ranges from -1 (perfect [[bimodal]]ity) to +1 (perfect [[unimodal]]ity). It is defined as

<math display="block"> A = U \left( 1 - \frac{ S - 1 }{ K - 1 } \right) </math>

where ''U'' is the unimodality of the distribution, ''S'' the number of categories that have nonzero frequencies and ''K'' the total number of categories.

The value of U is 1 if the distribution has any of the three following characteristics:

* all responses are in a single category
* the responses are evenly distributed among all the categories
* the responses are evenly distributed among two or more contiguous categories, with the other categories with zero responses

With distributions other than these the data must be divided into 'layers'. Within a layer the responses are either equal or zero. The categories do not have to be contiguous. A value for ''A'' for each layer (''A''<sub>i</sub>) is calculated and a weighted average for the distribution is determined. The weights (''w''<sub>i</sub>) for each layer are the number of responses in that layer. In symbols

<math display="block"> A_\text{overall} = \sum_i w_i A_i </math>

A [[Uniform distribution (discrete)|uniform distribution]] has ''A'' = 0: when all the responses fall into one category ''A'' = +1.

One theoretical problem with this index is that it assumes that the intervals are equally spaced. This may limit its applicability.

===Bimodal separation===

This index assumes that the distribution is a mixture of two normal distributions with means (''μ''<sub>1</sub> and ''μ''<sub>2</sub>) and standard deviations (''σ''<sub>1</sub> and ''σ''<sub>2</sub>):<ref name=Zhang2003>{{cite journal | last1 = Zhang | first1 = C | last2 = Mapes | first2 = BE | last3 = Soden | first3 = BJ | year = 2003 | title = Bimodality in tropical water vapour | journal = Quarterly Journal of the Royal Meteorological Society | volume = 129 | issue = 594| pages = 2847–2866 | doi =  10.1256/qj.02.166| bibcode = 2003QJRMS.129.2847Z | s2cid = 17153773 }}</ref>

<math display="block"> S = \frac{ \mu_1 - \mu_2 }{ 2( \sigma_1 +\sigma_2 ) } </math>

===Bimodality coefficient===

Sarle's bimodality coefficient ''b'' is<ref name=Ellison1987>{{cite journal | last1 = Ellison | first1 = AM | year = 1987 | title = Effect of seed dimorphism on the density-dependent dynamics of experimental populations of ''Atriplex triangularis'' (Chenopodiaceae) | journal = American Journal of Botany | volume = 74 | issue = 8| pages = 1280–1288 | doi=10.2307/2444163| jstor = 2444163 }}</ref>

<math display="block"> \beta = \frac{ \gamma^2 + 1 }{ \kappa }  </math>

where ''γ'' is the [[skewness]] and ''κ'' is the [[kurtosis]]. The kurtosis is here defined to be the standardised fourth moment around the mean. The value of ''b'' lies between 0 and 1.<ref name=Pearson1916>{{cite journal | last1 = Pearson | first1 = K | year = 1916 | title = Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew variation | journal = Philosophical Transactions of the Royal Society A | volume = 216 | issue = 538–548| pages = 429–457 | doi = 10.1098/rsta.1916.0009 | jstor = 91092 | bibcode =   1916RSPTA.216..429P| doi-access = free }}</ref> The logic behind this coefficient is that a bimodal distribution with light tails will have very low kurtosis, an asymmetric character, or both – all of which increase this coefficient.

The formula for a finite sample is<ref name=SASInst2012>SAS Institute Inc. (2012). SAS/STAT 12.1 user’s guide. Cary, NC: Author.</ref>

<math display="block"> b = \frac{ g^2 + 1 }{ k + \frac{ 3( n - 1 )^2 }{ ( n - 2 )( n - 3 ) } } </math>

where ''n'' is the number of items in the sample, ''g'' is the [[sample skewness]] and ''k'' is the sample [[excess kurtosis]].

The value of ''b'' for the [[uniform distribution (continuous)|uniform distribution]] is 5/9. This is also its value for the [[exponential distribution]]. Values greater than 5/9 may indicate a bimodal or multimodal distribution, though corresponding values can also result for heavily skewed unimodal distributions.<ref>{{cite journal | last1 = Pfister | first1 = R | last2 = Schwarz | first2 = KA | last3 = Janczyk | first3 = M. | last4 = Dale | first4 = R | last5 = Freeman | first5 = JB | year = 2013 | title = Good things peak in pairs: A note on the bimodality coefficient | journal =  Frontiers in Psychology| volume = 4 | pages = 700 | doi = 10.3389/fpsyg.2013.00700| pmid = 24109465 | pmc = 3791391 | doi-access = free }}</ref> The maximum value (1.0) is reached only by a [[Bernoulli distribution]] with only two distinct values or the sum of two different [[Dirac delta function]]s (a bi-delta distribution).

The distribution of this statistic is unknown. It is related to a statistic proposed earlier by Pearson – the difference between the kurtosis and the square of the skewness (''vide infra'').

===Bimodality amplitude===

This is defined as<ref name=Zhang2003/>

<math display="block"> A_B = \frac{A_1 - A_{ an } }{ A_1 } </math>

where ''A''<sub>1</sub> is the amplitude of the smaller peak and ''A''<sub>an</sub> is the amplitude of the antimode.

''A''<sub>B</sub> is always < 1. Larger values indicate more distinct peaks.

===Bimodal ratio===

This is the ratio of the left and right peaks.<ref name=Zhang2003/> Mathematically

<math display="block"> R = \frac{ A_r }{ A_l } </math>

where ''A''<sub>l</sub> and ''A''<sub>r</sub> are the amplitudes of the left and right peaks respectively.

===Bimodality parameter===

This parameter (''B'') is due to Wilcock.<ref name=Wilcock1993>{{cite journal | last1 = Wilcock | first1 = PR | year = 1993 | title = The critical shear stress of natural sediments | journal = Journal of Hydraulic Engineering | volume = 119 | issue = 4| pages = 491–505 | doi=10.1061/(asce)0733-9429(1993)119:4(491)}}</ref>

<math display="block"> B = \sqrt{ \frac{ A_r }{ A_l } } \sum_i P_i </math>

where ''A''<sub>l</sub> and ''A''<sub>r</sub> are the amplitudes of the left and right peaks respectively and ''P''<sub>''i''</sub> is the logarithm taken to the base 2 of the proportion of the distribution in the i<sup>th</sup> interval. The maximal value of the ''ΣP'' is 1 but the value of ''B'' may be greater than this.

To use this index, the log of the values are taken. The data is then divided into interval of width Φ whose value is log 2. The width of the peaks are taken to be four times 1/4Φ centered on their maximum values.

===Bimodality indices===

==== Wang's index ====

The bimodality index proposed by Wang ''et al'' assumes that the distribution is a sum of two normal distributions with equal variances but differing means.<ref name=Wang2009>{{cite journal | last1 = Wang | first1 = J | last2 = Wen | first2 = S | last3 = Symmans | first3 = WF | last4 = Pusztai | first4 = L | last5 = Coombes | first5 = KR | year = 2009 | title = The bimodality index: a criterion for discovering and ranking bimodal signatures from cancer gene expression profiling data | journal = Cancer Informatics | volume = 7 | pages = 199–216 |doi=10.4137/CIN.S2846| pmid = 19718451 | pmc = 2730180 }}</ref> It is defined as follows:

<math display="block"> \delta = \frac{ | \mu_1 - \mu_2 |}{ \sigma } </math>

where ''μ''<sub>1</sub>, ''μ''<sub>2</sub> are the means and ''σ'' is the common standard deviation.

<math display="block"> BI = \delta \sqrt{ p( 1 - p ) } </math>

where ''p'' is the mixing parameter.

==== Sturrock's index ====

A different bimodality index has been proposed by Sturrock.<ref name=Sturrock2008>{{cite journal | last1 = Sturrock | first1 = P | year = 2008 | title = Analysis of bimodality in histograms formed from GALLEX and GNO solar neutrino data | journal = Solar Physics | volume = 249 | issue = 1| pages = 1–10 | doi=10.1007/s11207-008-9170-3|arxiv = 0711.0216 |bibcode = 2008SoPh..249....1S | s2cid = 118389173 }}</ref>

This index (''B'') is defined as

<math display="block"> B = \frac{ 1 }{ N } \left[ \left( \sum_1^N \cos ( 2 \pi m \gamma ) \right)^2 +  \left( \sum_1^N \sin ( 2 \pi m \gamma ) \right)^2 \right] </math>

When ''m'' = 2 and ''γ'' is uniformly distributed, ''B'' is exponentially distributed.<ref name=Scargle1082>{{cite journal | last1 = Scargle | first1 = JD | year = 1982 | title = Studies in astronomical time series analysis. II – Statistical aspects of spectral analysis of unevenly spaced data | journal = The Astrophysical Journal | volume = 263 | issue = 1| pages = 835–853 | doi=10.1086/160554 | bibcode=1982ApJ...263..835S}}</ref>

This statistic is a form of [[periodogram]]. It suffers from the usual problems of estimation and spectral leakage common to this form of statistic.

==== de Michele and Accatino's index ====

Another bimodality index has been proposed by de Michele and Accatino.<ref name=deMichele2014>{{cite journal | last1 = De Michele | first1 = C | last2 = Accatino | first2 = F | year = 2014 | title = Tree cover bimodality in savannas and forests emerging from the switching between two fire dynamics | journal = PLoS One | volume =  9| issue = 3| pages =  e91195| doi = 10.1371/journal.pone.0091195 |bibcode = 2014PLoSO...991195D | pmid=24663432 | pmc=3963849| doi-access = free }}</ref> Their index (''B'') is

<math display="block"> B = | \mu - \mu_M | </math>

where ''μ'' is the arithmetic mean of the sample and

<math display="block"> \mu_M = \frac{ \sum_{ i = 1 }^L m_i x_i }{ \sum_{ i = 1 }^L m_i } </math>

where ''m''<sub>''i''</sub> is number of data points in the ''i''<sup>th</sup> bin, ''x''<sub>''i''</sub> is the center of the ''i''<sup>th</sup> bin and ''L'' is the number of bins.

The authors suggested a cut off value of 0.1 for ''B'' to distinguish between a bimodal (''B'' > 0.1)and unimodal (''B'' < 0.1) distribution. No statistical justification was offered for this value.

==== Sambrook Smith's index ====

A further index (''B'') has been proposed by Sambrook Smith ''et al''<ref name=SambrookSmith1997>{{cite journal | last1 = Sambrook Smith | first1 = GH | last2 = Nicholas | first2 = AP | last3 = Ferguson | first3 = RI | year = 1997 | title = Measuring and defining bimodal sediments: Problems and implications | journal = Water Resources Research | volume = 33 | issue = 5| pages = 1179–1185 | doi=10.1029/97wr00365 | bibcode=1997WRR....33.1179S| doi-access = free }}</ref>

<math display="block"> B = | \phi_2 - \phi_1 | \frac{ p_2 }{ p_1 } </math>

where ''p''<sub>1</sub> and ''p''<sub>2</sub> are the proportion contained in the primary (that with the greater amplitude) and secondary (that with the lesser amplitude) mode and ''φ''<sub>1</sub> and ''φ''<sub>2</sub> are the ''φ''-sizes of the primary and secondary mode. The ''φ''-size is defined as minus one times the log of the data size taken to the base 2. This transformation is commonly used in the study of sediments.

The authors recommended a cut off value of 1.5 with B being greater than 1.5 for a bimodal distribution and less than 1.5 for a unimodal distribution. No statistical justification for this value was given.

==== Otsu's method ====

[[Otsu's method]] for finding a threshold for separation between two modes relies on minimizing the quantity
<math display=block> \frac{ n_1 \sigma_1^2 + n_2 \sigma_2^2 }{ m \sigma^2 } </math>
where ''n''<sub>''i''</sub> is the number of data points in the ''i''<sup>th</sup> subpopulation, ''σ''<sub>''i''</sub><sup>2</sup> is the variance of the ''i''<sup>th</sup> subpopulation, ''m'' is the total size of the sample and ''σ''<sup>2</sup> is the sample variance. Some researchers (particularly in the field of [[digital image processing]]) have applied this quantity more broadly as an index for detecting bimodality, with a small value indicating a more bimodal distribution.<ref name=Chaudhuri2010>{{cite journal | last1 = Chaudhuri | first1 = D | last2 = Agrawal | first2 = A | year = 2010 | title = Split-and-merge procedure for image segmentation using bimodality detection approach | journal = Defence Science Journal | volume = 60 | issue = 3| pages = 290–301 | doi=10.14429/dsj.60.356| doi-access =  }}</ref>

==Statistical tests==

A number of tests are available to determine if a data set is distributed in a bimodal (or multimodal) fashion.

===Graphical methods===

In the study of sediments, particle size is frequently bimodal. Empirically, it has been found useful to plot the frequency against the log(&nbsp;size&nbsp;) of the particles.<ref name=Folk1957>{{cite journal | last1 = Folk | first1 = RL | last2 = Ward | first2 = WC | year = 1957 | title = Brazos River bar: a study in the significance of grain size parameters | url = https://doi.pangaea.de/10.1594/PANGAEA.896129| journal = Journal of Sedimentary Research | volume = 27 | issue = 1| pages = 3–26 | doi=10.1306/74d70646-2b21-11d7-8648000102c1865d|bibcode = 1957JSedR..27....3F }}</ref><ref name=Dyer1970>{{cite journal | last1 = Dyer | first1 = KR | year = 1970 | title = Grain-size parameters for sandy gravels | journal = Journal of Sedimentary Research | volume = 40 | issue = 2| pages = 616–620 |doi=10.1306/74D71FE6-2B21-11D7-8648000102C1865D}}</ref> This usually gives a clear separation of the particles into a bimodal distribution. In geological applications the [[logarithm]] is normally taken to the base 2. The log transformed values are referred to as phi (Φ) units. This system is known as the [[Grain size|Krumbein]] (or phi) scale.

An alternative method is to plot the log of the particle size against the cumulative frequency. This graph will usually consist two reasonably straight lines with a connecting line corresponding to the antimode.

;Statistics

Approximate values for several statistics can be derived from the graphic plots.<ref name=Folk1957/>

<math display="block">\begin{align}
\text{mean} &= \frac{ \phi_{16} + \phi_{50} + \phi_{84} }{ 3 } \\[1ex]
\text{std. dev.} &= \frac{ \phi_{84} - \phi_{16} }{ 4 } + \frac{ \phi_{95} - \phi_5 }{ 6.6 } \\[1ex]
\text{skewness} &= \frac{ \phi_{84} +  \phi_{16} - 2  \phi_{50} }{ 2 ( \phi_{84} -  \phi_{16} ) } + \frac{ \phi_{95} +  \phi_{ 5 } -  2 \phi_{50} }{ 2( \phi_{95} - \phi_5 ) } \\[1ex]
\text{kurtosis} &= \frac{ \phi_{95} - \phi_5 }{ 2.44 ( \phi_{75} - \phi_{25} ) }
\end{align}</math>

where ''φ''<sub>x</sub> is the value of the variate ''φ'' at the ''x''<sup>th</sup> percentage of the distribution.

===Unimodal vs. bimodal distribution===

Pearson in 1894 was the first to devise a procedure to test whether a distribution could be resolved into two normal distributions.<ref name=Pearson1894>{{cite journal | last1 = Pearson | first1 = K | year = 1894 | title = Contributions to the mathematical theory of evolution: On the dissection of asymmetrical frequency-curves | journal = Philosophical Transactions of the Royal Society A | volume = 185 | pages = 71–90 | doi=10.1098/rsta.1894.0003| bibcode = 1894RSPTA.185...71P| doi-access = free }}</ref> This method required the solution of a ninth order [[polynomial]]. In a subsequent paper Pearson reported that for any distribution skewness<sup>2</sup> + 1 < kurtosis.<ref name=Pearson1916/> Later Pearson showed that<ref name=Pearson1929>{{cite journal | last1 = Pearson | first1 = K | year = 1929 | title = Editorial note | journal = Biometrika | volume = 21 | pages = 370–375 }}</ref>

<math display="block"> b_2 - b_1 \ge 1 </math>

where ''b''<sub>2</sub> is the kurtosis and ''b''<sub>1</sub> is the square of the skewness. Equality holds only for the two point [[Bernoulli distribution]] or the sum of two different [[Dirac delta function]]s. These are the most extreme cases of bimodality possible. The kurtosis in both these cases is 1. Since they are both symmetrical their skewness is 0 and the difference is 1.

Baker proposed a transformation to convert a bimodal to a unimodal distribution.<ref name=Baker1930>{{cite journal | last1 = Baker | first1 = GA | year = 1930 | title = Transformations of bimodal distributions | journal = Annals of Mathematical Statistics | volume = 1 | issue = 4| pages = 334–344 | doi=10.1214/aoms/1177733063| doi-access = free }}</ref>

Several tests of unimodality versus bimodality have been proposed: Haldane suggested one based on second central differences.<ref name=Haldane1951>{{cite journal | last1 = Haldane | first1 = JBS | year = 1951 | title = Simple tests for bimodality and bitangentiality | journal = Annals of Eugenics  | volume = 16 | issue = 1| pages = 359–364 | doi = 10.1111/j.1469-1809.1951.tb02488.x | pmid = 14953132 }}</ref> Larkin later introduced a test based on the F test;<ref name=Larkin1979>{{cite journal | last1 = Larkin | first1 = RP | year = 1979 | title = An algorithm for assessing bimodality vs. unimodality in a univariate distribution | journal = Behavior Research Methods & Instrumentation | volume = 11 | issue = 4| pages = 467–468 | doi = 10.3758/BF03205709 | doi-access = free }}</ref> Benett created one based on [[G-test|Fisher's G test]].<ref name=Bennett1992>{{cite journal | last1 = Bennett | first1 = SC | year = 1992 | title = Sexual dimorphism of ''Pteranodon'' and other pterosaurs, with comments on cranial crests | journal = Journal of Vertebrate Paleontology | volume = 12 | issue = 4| pages = 422–434 | doi=10.1080/02724634.1992.10011472}}</ref> Tokeshi has proposed a fourth test.<ref name=Tokeshi1992>{{cite journal | last1 = Tokeshi | first1 = M | year = 1992 | title = Dynamics and distribution in animal communities; theory and analysis | journal = Researches on Population Ecology | volume = 34 | issue = 2| pages = 249–273 | doi=10.1007/bf02514796| s2cid = 22912914 }}</ref><ref name=Barreto2003>{{cite journal | last1 = Barreto | first1 = S | last2 = Borges | first2 = PAV | last3 = Guo | first3 = Q | year = 2003 | title = A typing error in Tokeshi's test of bimodality | journal = Global Ecology and Biogeography | volume = 12 | issue = 2| pages = 173–174 | doi=10.1046/j.1466-822x.2003.00018.x| hdl = 10400.3/1408 | hdl-access = free }}</ref> A test based on a likelihood ratio has been proposed by Holzmann and Vollmer.<ref name=Holzmann2008/>

A method based on the score and Wald tests has been proposed.<ref name=Carolan2001>{{cite journal | last1 = Carolan | first1 = AM | last2 = Rayner | first2 = JCW | year = 2001 | title = One sample tests for the location of modes of nonnormal data | journal =  Journal of Applied Mathematics and Decision Sciences| volume = 5 | issue = 1| pages = 1–19 | doi=10.1155/s1173912601000013| citeseerx = 10.1.1.504.4999 | doi-access = free }}</ref> This method can distinguish between unimodal and bimodal distributions when the underlying distributions are known.

===Antimode tests===

Statistical tests for the antimode are known.<ref name=Hartigan2000>{{cite book |last=Hartigan |first=J. A. |date=2000 |chapter=Testing for Antimodes |editor1=Gaul W |editor2=Opitz O |editor3=Schader M |title=Data Analysis |series=Studies in Classification, Data Analysis, and Knowledge Organization |publisher=Springer |pages=169–181 |isbn=3-540-67731-3 |chapter-url=https://books.google.com/books?id=WVDmCAAAQBAJ&pg=PA169 }}</ref>

;Otsu's method

[[Otsu's method]] is commonly employed in computer graphics to determine the optimal separation between two distributions.

===General tests===

To test if a distribution is other than unimodal, several additional tests have been devised: the [[bandwidth test (multimodal)|bandwidth test]],<ref name=Silverman1981/> the [[dip test]],<ref name=Hartigan1985>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Hartigan | first2 = PM | year = 1985 | title = The dip test of unimodality | journal = Annals of Statistics | volume = 13 | issue = 1| pages = 70–84 | doi=10.1214/aos/1176346577| doi-access = free }}</ref> the [[excess mass test]],<ref name=Mueller1991>{{cite journal | last1 = Mueller | first1 = DW | last2 = Sawitzki | first2 = G | year = 1991 | title = Excess mass estimates and tests for multimodality | journal = Journal of the American Statistical Association | volume = 86 | issue = 415| pages = 738–746 |jstor=2290406 | doi=10.1080/01621459.1991.10475103}}</ref> the MAP test,<ref name="Rozál1994">{{cite journal | last1 = Rozál | first1 = GPM Hartigan JA | year = 1994 | title = The MAP test for multimodality | journal = Journal of Classification | volume = 11 | issue = 1| pages = 5–36 | doi = 10.1007/BF01201021 | s2cid = 118500771 }}</ref> the [[mode existence test]],<ref name=Minnotte1997>{{cite journal | last1 = Minnotte | first1 = MC | year = 1997 | title = Nonparametric testing of the existence of modes | journal = Annals of Statistics | volume = 25 | issue = 4| pages = 1646–1660 | doi=10.1214/aos/1031594735| doi-access = free }}</ref> the [[runt test]],<ref name=Hartigan1992>{{cite journal | last1 = Hartigan | first1 = JA | last2 = Mohanty | first2 = S | year = 1992 | title = The RUNT test for multimodality | journal = Journal of Classification | volume = 9 | pages = 63–70 | doi=10.1007/bf02618468| s2cid = 121960832 }}</ref><ref name=Andrushkiw2008>{{cite journal |author1=Andrushkiw RI |author2=Klyushin DD |author3=Petunin YI |date=2008 |title=A new test for unimodality |journal=Theory of Stochastic Processes |volume=14 |issue=1 |pages=1–6}}</ref> the [[span test]],<ref name=Hartigan1988>{{cite book |last=Hartigan |first=J. A. |year=1988 |chapter=The Span Test of Multimodality |title=Classification and Related Methods of Data Analysis |editor-first=H. H. |editor-last=Bock |publisher=North-Holland |location=Amsterdam |pages=229–236 |isbn=0-444-70404-3 }}</ref> and the [[saddle test]].

An implementation of the dip test is available for the [[R (programming language)|R programming language]].<ref>{{cite web|url=https://cran.r-project.org/web/packages/diptest/index.html|title=diptest: Hartigan's Dip Test Statistic for Unimodality - Corrected|first1=Martin Maechler (originally from Fortran and S.-plus by Dario|last1=Ringach|last2=NYU.edu)|date=5 December 2016|via=R-Packages}}</ref> The p-values for the dip statistic values range between 0 and 1. P-values less than 0.05 indicate significant multimodality and p-values greater than 0.05 but less than 0.10 suggest multimodality with marginal significance.<ref name=FreemanDale2012>{{cite journal | last1 = Freeman | last2 = Dale | year = 2012 | title = Assessing bimodality to detect the presence of a dual cognitive process | journal = Behavior Research Methods | volume = 45 | issue = 1 | pages = 83–97 | doi = 10.3758/s13428-012-0225-x | pmid = 22806703 | s2cid = 14500508 | url = http://psych.nyu.edu/freemanlab/pubs/2012_BRM.pdf| doi-access = free }}</ref>

===Silverman's test===

Silverman introduced a bootstrap method for the number of modes.<ref name=Silverman1981>{{cite journal | last1 = Silverman | first1 = B. W. | year = 1981 | title = Using kernel density estimates to investigate multimodality | journal = Journal of the Royal Statistical Society, Series B | volume = 43 | issue = 1| pages = 97–99 |jstor=2985156| bibcode = 1981JRSSB..43...97S | doi=10.1111/j.2517-6161.1981.tb01155.x}}</ref> The test uses a fixed bandwidth which reduces the power of the test and its interpretability. Under smoothed densities may have an excessive number of modes whose count during bootstrapping is unstable.

===Bajgier-Aggarwal test===

Bajgier and Aggarwal have proposed a test based on the kurtosis of the distribution.<ref name=Bajgier1991>{{cite journal |author1=Bajgier SM |author2=Aggarwal LK |date=1991 |title=Powers of goodness-of-fit tests in detecting balanced mixed normal distributions |journal=Educational and Psychological Measurement |volume=51 |issue=2 |pages=253–269 |doi=10.1177/0013164491512001|s2cid=121113601 }}</ref>

===Special cases===

Additional tests are available for a number of special cases:

;Mixture of two normal distributions

A study of a mixture density of two normal distributions data found that separation into the two normal distributions was difficult unless the means were separated by 4–6 standard deviations.<ref name=Jackson1898>{{cite journal | last1 = Jackson | first1 = PR | last2 = Tucker | first2 = GT | last3 = Woods | first3 = HF | year = 1989 | title = Testing for bimodality in frequency distributions of data suggesting polymorphisms of drug metabolism--hypothesis testing | journal = British Journal of Clinical Pharmacology | volume = 28 | issue = 6| pages = 655–662 | doi=10.1111/j.1365-2125.1989.tb03558.x| pmid = 2611088 | pmc = 1380036 }}</ref>

In [[astronomy]] the Kernel Mean Matching algorithm is used to decide if a data set belongs to a single normal distribution or to a mixture of two normal distributions.

;Beta-normal distribution

This distribution is bimodal for certain values of is parameters. A test for these values has been described.<ref>{{cite conference|url=http://www.amstat.org/sections/srms/Proceedings/y2002/Files/JSM2002-000150.pdf|archive-url=https://web.archive.org/web/20160304051936/http://www.amstat.org/sections/srms/Proceedings/y2002/Files/JSM2002-000150.pdf|archive-date=2016-03-04|contribution=Beta-normal distribution: Bimodality properties and application|first1=Felix|last1=Famoye|first2=Carl|last2=Lee|first3=Nicholas|last3=Eugene|title=Joint Statistical Meetings - Section on Physical & Engineering Sciences (SPES)|publisher=American Statistical Society|pages=951-956}}</ref>

==Parameter estimation and fitting curves==

Assuming that the distribution is known to be bimodal or has been shown to be bimodal by one or more of the tests above, it is frequently desirable to fit a curve to the data. This may be difficult.

Bayesian methods may be useful in difficult cases.

===Software===

;Two normal distributions

A package for [[R (programming language)|R]] is available for testing for bimodality.<ref>{{cite web |url=http://www.uni-marburg.de/fb12/stoch/research/rpackage/manualbimodlilitytest.pdf |title=Archived copy |access-date=2013-11-01 |url-status=dead |archive-url=https://web.archive.org/web/20131103100209/http://www.uni-marburg.de/fb12/stoch/research/rpackage/manualbimodlilitytest.pdf |archive-date=2013-11-03 }}</ref> This package assumes that the data are distributed as a sum of two normal distributions. If this assumption is not correct the results may not be reliable. It also includes functions for fitting a sum of two normal distributions to the data.

Assuming that the distribution is a mixture of two normal distributions then the expectation-maximization algorithm may be used to determine the parameters. Several programmes are available for this including Cluster,<ref>{{cite web|url=https://engineering.purdue.edu/~bouman/software/cluster/|title=Cluster home page|website=engineering.purdue.edu}}</ref> and the R package nor1mix.<ref>{{cite web|url=https://cran.r-project.org/web/packages/nor1mix/index.html|title=nor1mix: Normal (1-d) Mixture Models (S3 Classes and Methods)|first=Martin|last=Mächler|date=25 August 2016|via=R-Packages}}</ref>

;Other distributions

The mixtools package available for R can test for and estimate the parameters of a number of different distributions.<ref>{{cite web|url=https://cran.r-project.org/web/packages/mixtools/index.html|title=mixtools: Tools for Analyzing Finite Mixture Models|first1=Derek|last1=Young|first2=Tatiana|last2=Benaglia|first3=Didier|last3=Chauveau|first4=David|last4=Hunter|first5=Ryan|last5=Elmore|first6=Thomas|last6=Hettmansperger|first7=Hoben|last7=Thomas|first8=Fengjuan|last8=Xuan|date=10 March 2017|via=R-Packages}}</ref> A package for a mixture of two right-tailed gamma distributions is available.<ref>{{cite web|title=discrimARTs|url=https://cran.r-project.org/web/packages/discrimARTs/discrimARTs.pdf|website=cran.r-project.org|access-date=22 March 2018}}</ref>

Several other packages for R are available to fit mixture models; these include flexmix,<ref>{{cite web|url=https://cran.r-project.org/web/packages/flexmix/index.html|title=flexmix: Flexible Mixture Modeling|first1=Bettina|last1=Gruen|first2=Friedrich|last2=Leisch|first3=Deepayan|last3=Sarkar|first4=Frederic|last4=Mortier|first5=Nicolas|last5=Picard|date=28 April 2017|via=R-Packages}}</ref> mcclust,<ref>{{cite web|url=https://cran.r-project.org/web/packages/mclust/index.html|title=mclust: Gaussian Mixture Modelling for Model-Based Clustering, Classification, and Density Estimation|first1=Chris|last1=Fraley|first2=Adrian E.|last2=Raftery|first3=Luca|last3=Scrucca|first4=Thomas Brendan|last4=Murphy|first5=Michael|last5=Fop|date=21 May 2017|via=R-Packages}}</ref> agrmt,<ref>{{cite web|url=https://cran.r-project.org/web/packages/agrmt/index.html|title=agrmt|first1=Didier|last1=Ruedin|date=2 April 2016|publisher=cran.r-project.org}}</ref> and mixdist.<ref>{{cite web|url=https://cran.r-project.org/web/packages/mixdist/index.html|title=mixdist: Finite Mixture Distribution Models|first1=Peter|last1=Macdonald|first2=with contributions from Juan|last2=Du|date=29 October 2012|via=R-Packages}}</ref>

The statistical programming language [[SAS language|SAS]] can also fit a variety of mixed distributions with the PROC FREQ procedure.
[[File:Joggers.png|thumb|Number of joggers in a park by time of the day (X in hours) in a bimodal probability distribution]]

In Python, the package [[Scikit-learn]] contains a tool for mixture modeling<ref>{{cite web|title=Gaussian mixture models|url=https://scikit-learn.org/stable/modules/mixture.html#mixture|website=scikit-learn.org|access-date=30 November 2023}}</ref>

===Example software application===
The CumFreqA <ref>CumFreq, free program for fitting of probability distributions to a data set. On line: [https://www.waterlog.info/cumfreq.htm]</ref> program for the fitting of composite probability distributions to a data set (X) can divide the set into two parts with a different distribution. The figure shows an example of a double generalized mirrored [[Gumbel distribution]] as in [[distribution fitting]] with cumulative distribution function (CDF) equations:

 X < 8.10 : CDF = 1 - exp[-exp{-(0.092X'''^'''0.01+935)}]
 X > 8.10 : CDF = 1 - exp[-exp{-(-0.0039X'''^'''2.79+1.05)}]

==See also==
* [[Overdispersion]]
* [[Mixture model]] - Gaussian Mixture Models (GMM)
* [[Mixture distribution]]

==References==
{{reflist|30em}}

{{ProbDistributions}}

[[Category:Continuous distributions]]