Editing Coefficient of variation (section)

==Estimation==
When only a sample of data from a population is available, the population CV can be estimated using the ratio of the [[Standard deviation#Estimation|sample standard deviation]] <math>s \,</math> to the sample mean <math>\bar{x}</math>:

:<math>\widehat{c_{\rm v}} = \frac{s}{\bar{x}}</math>

But this estimator, when applied to a small or moderately sized sample, tends to be too low: it is a [[biased estimator]].  For [[normally distributed]] data, an unbiased estimator<ref>Sokal RR & Rohlf FJ. ''Biometry'' (3rd Ed). New York: Freeman, 1995. p. 58. {{ISBN|0-7167-2411-1}}</ref> for a sample of size n is:

:<math>\widehat{c_{\rm v}}^*=\bigg(1+\frac{1}{4n}\bigg)\widehat{c_{\rm v}}</math>

===Log-normal data===
Many datasets follow an approximately log-normal distribution.<ref>{{cite journal |doi=10.1641/0006-3568(2001)051[0341:LNDATS]2.0.CO;2 |title=Log-normal Distributions across the Sciences: Keys and Clues |year=2001 |last1=Limpert |first1=Eckhard |last2=Stahel |first2=Werner A. |last3=Abbt |first3=Markus |journal=BioScience |volume=51 |issue=5 |pages=341–352|doi-access=free }}</ref> In such cases, a more accurate estimate, derived from the properties of the [[log-normal distribution]],<ref>{{cite journal |doi=10.1093/biomet/51.1-2.25 |title=Confidence intervals for the coefficient of variation for the normal and log normal distributions |year=1964 |last1=Koopmans |first1=L. H. |last2=Owen |first2=D. B. |last3=Rosenblatt |first3=J. I. |journal=Biometrika |volume=51 |issue=1–2 |pages=25–32}}</ref><ref>{{cite journal |pmid=1601532 |year=1992 |last1=Diletti |first1=E |last2=Hauschke |first2=D |last3=Steinijans |first3=VW |title=Sample size determination for bioequivalence assessment by means of confidence intervals |volume=30 |pages=S51–8 |journal=International Journal of Clinical Pharmacology, Therapy, and Toxicology|issue=Suppl 1 }}</ref><ref>{{cite journal |doi=10.1081/BIP-100101013 |title=Why Are Pharmacokinetic Data Summarized by Arithmetic Means? |year=2000 |last1=Julious |first1=Steven A. |last2=Debarnot |first2=Camille A. M. |journal=Journal of Biopharmaceutical Statistics |volume=10 |pages=55–71 |pmid=10709801 |issue=1|s2cid=2805094 }}</ref> is defined as:

:<math>\widehat{cv}_{\rm raw} = \sqrt{\mathrm{e}^{s_{\ln}^2}-1}</math>

where <math>{s_{\ln}} \,</math> is the sample standard deviation of the data after a [[natural log]] transformation.  (In the event that measurements are recorded using any other logarithmic base, b, their standard deviation <math>s_b \,</math> is converted to base e using <math>s_{\ln} = s_b \ln(b) \,</math>, and the formula for <math>\widehat{cv}_{\rm raw} \,</math> remains the same.<ref>{{cite journal | last1 = Reed | first1 = JF | last2 = Lynn | first2 = F | last3 = Meade | first3 = BD | year = 2002 | title = Use of Coefficient of Variation in Assessing Variability of Quantitative Assays | journal = Clin Diagn Lab Immunol | volume = 9 | issue = 6| pages = 1235–1239 | doi = 10.1128/CDLI.9.6.1235-1239.2002 | pmid = 12414755 | pmc = 130103 }}</ref>)  This estimate is sometimes referred to as the "geometric CV" (GCV)<ref>Sawant, S.; Mohan, N. (2011) [http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS"] {{webarchive|url=https://web.archive.org/web/20110824094357/http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf |date=24 August 2011 }}, ''PharmaSUG2011'', Paper PO08</ref><ref>{{cite journal | last1 = Schiff | first1 = MH | display-authors = etal   | year = 2014 | title =  Head-to-head, randomised, crossover study of oral versus subcutaneous methotrexate in patients with rheumatoid arthritis: drug-exposure limitations of oral methotrexate at doses >=15 mg may be overcome with subcutaneous administration| journal = Ann Rheum Dis | volume = 73| issue = 8| pages = 1–3 | doi = 10.1136/annrheumdis-2014-205228 | pmid = 24728329 | pmc = 4112421}}</ref> in order to distinguish it from the simple estimate above.  However, "geometric coefficient of variation" has also been defined by Kirkwood<ref>{{cite journal |last1=Kirkwood |first1=TBL |title=Geometric means and measures of dispersion |journal=Biometrics |year=1979 |volume=35 |issue=4 |pages=908–9 |jstor=2530139 }}</ref> as:

:<math>\mathrm{GCV_K} = {\mathrm{e}^{s_{\ln}}\!\!-1}</math>

This term was intended to be ''analogous'' to the coefficient of variation, for describing multiplicative variation in log-normal data, but this definition of GCV has no theoretical basis as an estimate of <math>c_{\rm v} \,</math> itself.

For many practical purposes (such as [[sample size determination]] and calculation of [[confidence intervals]]) it is <math>s_{ln} \,</math> which is of most use in the context of log-normally distributed data.  If necessary, this can be derived from an estimate of <math>c_{\rm v} \,</math> or GCV by inverting the corresponding formula.