Editing Coefficient of variation (section)

==Applications==
The coefficient of variation is also common in applied probability fields such as [[renewal theory]], [[queueing theory]], and [[reliability theory]].  In these fields, the [[exponential distribution]] is often more important than the [[normal distribution]].
The standard deviation of an [[exponential distribution]] is equal to its mean, so its coefficient of variation is equal to 1.  Distributions with CV < 1 (such as an [[Erlang distribution]]) are considered low-variance, while those with CV > 1 (such as a [[hyper-exponential distribution]]) are considered high-variance{{Citation needed|date=June 2019}}.  Some formulas in these fields are expressed using the '''squared coefficient of variation''', often abbreviated SCV. In modeling, a variation of the CV is the CV(RMSD).  Essentially the CV(RMSD) replaces the standard deviation term with the [[RMSD|Root Mean Square Deviation (RMSD)]]. While many natural processes indeed show a correlation between the average value and the amount of variation around it, accurate sensor devices need to be designed in such a way that the coefficient of variation is close to zero, i.e., yielding a constant [[absolute error]] over their working range.

In [[actuarial science]], the CV is known as '''unitized risk'''.<ref>{{cite book|last1=Broverman|first1=Samuel A.|title=Actex study manual, Course 1, Examination of the Society of Actuaries, Exam 1 of the Casualty Actuarial Society|date=2001|publisher=Actex Publications|location=Winsted, CT|isbn=9781566983969|page=104|edition=2001|url=https://books.google.com/books?id=qd4PAQAAMAAJ&q=%22unitized+risk%22|access-date=7 June 2014}}</ref>

In industrial solids processing, CV is particularly important to measure the degree of homogeneity of a powder mixture. Comparing the calculated CV to a specification will allow to define if a sufficient degree of mixing has been reached.<ref>{{cite web|url=https://www.powderprocess.net/Measuring_Degree_Mixing.html|title=Measuring Degree of Mixing – Homogeneity of powder mix - Mixture quality - PowderProcess.net|website=www.powderprocess.net|access-date=2 May 2018|url-status=live|archive-url=https://web.archive.org/web/20171114145327/https://www.powderprocess.net/Measuring_Degree_Mixing.html|archive-date=14 November 2017|df=dmy-all}}</ref>

In [[fluid dynamics]], the '''CV''', also referred to as '''Percent RMS''', '''%RMS''', '''%RMS Uniformity''', or '''Velocity RMS''', is a useful determination of flow uniformity for industrial processes.  The term is used widely in the design of pollution control equipment, such as electrostatic precipitators (ESPs),<ref>{{cite web |url=http://www.airflowsciences.com/sites/default/files/docs/201810Int-ESP-Improved-Methodology-for-Accurate-CFD-and-Physical-Modeling-of-ESPs.pdf| title = Improved Methodology for Accurate CFD and Physical Modeling of ESPs | last1 = Banka | first1 = A | last2 = Dumont | first2 = B | last3 = Franklin | first3 = J | last4 = Klemm | first4 = G | last5 = Mudry | first5 = R | date = 2018 | publisher = International Society of Electrostatic Precipitation (ISESP) Conference 2018 }}</ref> selective catalytic reduction (SCR), scrubbers, and similar devices.  The Institute of Clean Air Companies (ICAC) references RMS deviation of velocity in the design of fabric filters (ICAC document F-7).<ref>{{cite web |url=https://www.icac.com/resource/resmgr/ICAC_F-7_FF_Flow_Model_Studi.pdf| title = F7 - Fabric Filter Gas Flow Model Studies | date = 1996 | publisher = Institute of Clean Air Companies (ICAC) }}</ref>  The guiding principle is that many of these pollution control devices require "uniform flow" entering and through the control zone.  This can be related to uniformity of velocity profile, temperature distribution, gas species (such as ammonia for an SCR, or activated carbon injection for mercury absorption), and other flow-related parameters.  The '''Percent RMS''' also is used to assess flow uniformity in combustion systems, HVAC systems, ductwork, inlets to fans and filters, air handling units, etc. where performance of the equipment is influenced by the incoming flow distribution.

===Laboratory measures of intra-assay and inter-assay CVs===
CV measures are often used as quality controls for quantitative laboratory [[assay]]s. While intra-assay and inter-assay CVs might be assumed to be calculated by simply averaging CV values across CV values for multiple samples within one assay or by averaging multiple inter-assay CV estimates, it has been suggested that these practices are incorrect and that a more complex computational process is required.<ref>{{cite journal|last=Rodbard|first=D|title=Statistical quality control and routine data processing for radioimmunoassays and immunoradiometric assays.|journal=Clinical Chemistry|date=October 1974|volume=20|issue=10|pages=1255–70|doi=10.1093/clinchem/20.10.1255|pmid=4370388|doi-access=free}}</ref> It has also been noted that CV values are not an ideal index of the certainty of a measurement when the number of replicates varies across samples − in this case standard error in percent is suggested to be superior.<ref name="ReferenceA">{{cite journal|last1=Eisenberg|first1=Dan|title=Improving qPCR telomere length assays: Controlling for well position effects increases statistical power|journal=American Journal of Human Biology|date=2015|doi=10.1002/ajhb.22690|pmid=25757675|volume=27|issue=4|pages=570–5|pmc=4478151}}</ref> If measurements do not have a natural zero point then the CV is not a valid measurement and alternative measures such as the [[intraclass correlation]] coefficient are recommended.<ref name="Eisenberg-CV-ICC">{{cite journal|last1=Eisenberg|first1=Dan T. A.|title=Telomere length measurement validity: the coefficient of variation is invalid and cannot be used to compare quantitative polymerase chain reaction and Southern blot telomere length measurement technique|journal=International Journal of Epidemiology|volume=45|issue=4|date=30 August 2016|pages=1295–1298|doi=10.1093/ije/dyw191|pmid=27581804|issn=0300-5771|doi-access=free}}</ref>

=== As a measure of economic inequality ===

The coefficient of variation fulfills the [[Income inequality metrics|requirements for a measure of economic inequality]].<ref name="Champ1999">{{cite book |last1=Champernowne |first1=D. G.|last2=Cowell |first2=F. A.|date=1999 |title=Economic Inequality and Income Distribution |publisher=Cambridge University Press }}</ref><ref name="Campano2006">{{cite book |last1=Campano |first1=F. |last2=Salvatore |first2=D. |date=2006 |title=Income distribution |publisher=Oxford University Press }}</ref><ref name="Bellu2006">{{cite web |url=http://www.fao.org/docs/up/easypol/448/simple_inequality_mesures_080en.pdf |title=Policy Impacts on Inequality – Simple Inequality Measures |last1=Bellu |first1=Lorenzo Giovanni |last2=Liberati |first2=Paolo |date=2006 |website=EASYPol, Analytical tools |publisher=Policy Support Service, Policy Assistance Division, FAO |access-date=13 June 2016 |url-status=live |archive-url=https://web.archive.org/web/20160805101141/http://www.fao.org/docs/up/easypol/448/simple_inequality_mesures_080en.pdf |archive-date=5 August 2016 |df=dmy-all }}</ref> If '''x''' (with entries ''x''<sub>''i''</sub>) is a list of the values of an economic indicator (e.g. wealth), with ''x''<sub>''i''</sub> being the wealth of agent ''i'', then the following requirements are met:

* Anonymity – ''c''<sub>''v''</sub> is independent of the ordering of the list '''x'''. This follows from the fact that the variance and mean are independent of the ordering of '''x'''.
* Scale invariance: ''c''<sub>v</sub>('''x''') = ''c''<sub>v</sub>(&alpha;'''x''') where ''&alpha;'' is a real number.<ref name="Bellu2006" />
* Population independence – If {'''x''','''x'''} is the list '''x''' appended to itself, then ''c''<sub>''v''</sub>({'''x''','''x'''}) = ''c''<sub>''v''</sub>('''x'''). This follows from the fact that the variance and mean both obey this principle.
* Pigou–Dalton transfer principle: when wealth is transferred from a wealthier agent ''i'' to a poorer agent ''j'' (i.e. ''x''<sub>''i''</sub>&nbsp;>&nbsp;''x''<sub>''j''</sub>) without altering their rank, then ''c''<sub>''v''</sub> decreases and vice versa.<ref name="Bellu2006" />

''c''<sub>''v''</sub> assumes its minimum value of zero for complete equality (all ''x''<sub>''i''</sub> are equal).<ref name="Bellu2006" /> Its most notable drawback is that it is not bounded from above, so it cannot be normalized to be within a fixed range (e.g. like the [[Gini coefficient]] which is constrained to be between 0 and 1).<ref name="Bellu2006" /> It is, however, more mathematically tractable than the Gini coefficient.

=== As a measure of standardisation of archaeological artefacts ===

Archaeologists often use CV values to compare the degree of standardisation of ancient artefacts.<ref>{{cite journal |last1=Eerkens |first1=Jelmer W. |last2=Bettinger |first2=Robert L. |title=Techniques for Assessing Standardization in Artifact Assemblages: Can We Scale Material Variability? |journal=American Antiquity |date=July 2001 |volume=66 |issue=3 |pages=493–504 |doi=10.2307/2694247|jstor=2694247 |s2cid=163507589 }}</ref><ref>{{cite journal |last1=Roux |first1=Valentine |title=Ceramic Standardization and Intensity of Production: Quantifying Degrees of Specialization |journal=American Antiquity |date=2003 |volume=68 |issue=4 |pages=768–782 |doi=10.2307/3557072 |jstor=3557072 |s2cid=147444325 |url=http://doi.org/10.2307/3557072 |language=en |issn=0002-7316}}</ref> Variation in CVs has been interpreted to indicate different cultural transmission contexts for the adoption of new technologies.<ref>{{cite journal |last1=Bettinger |first1=Robert L. |last2=Eerkens |first2=Jelmer |title=Point Typologies, Cultural Transmission, and the Spread of Bow-and-Arrow Technology in the Prehistoric Great Basin |journal=American Antiquity |date=April 1999 |volume=64 |issue=2 |pages=231–242 |doi=10.2307/2694276|jstor=2694276 |s2cid=163198451 }}</ref> Coefficients of variation have also been used to investigate pottery standardisation relating to changes in social organisation.<ref>{{cite journal |last1=Wang |first1=Li-Ying |last2=Marwick |first2=Ben |title=Standardization of ceramic shape: A case study of Iron Age pottery from northeastern Taiwan |journal=Journal of Archaeological Science: Reports |date=October 2020 |volume=33 |pages=102554 |doi=10.1016/j.jasrep.2020.102554|bibcode=2020JArSR..33j2554W |s2cid=224904703 |url=http://osf.io/q8hn9/ }}</ref> Archaeologists also use several methods for comparing CV values, for example the modified signed-likelihood ratio (MSLR) test for equality of CVs.<ref>{{cite journal |last1=Krishnamoorthy |first1=K. |last2=Lee |first2=Meesook |title=Improved tests for the equality of normal coefficients of variation |journal=Computational Statistics |date=February 2014 |volume=29 |issue=1–2 |pages=215–232 |doi=10.1007/s00180-013-0445-2|s2cid=120898013 }}</ref><ref>{{cite book |last1=Marwick |first1=Ben |last2=Krishnamoorthy |first2=K |title=cvequality: Tests for the equality of coefficients of variation from multiple groups |date=2019 |publisher=R package version 0.2.0. |url=https://cran.r-project.org/package=cvequality}}</ref>