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Coefficient of variation
It has been suggested that Relative standard deviation be merged into this article. (Discuss) Proposed since April 2013. 
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (November 2010) 
In probability theory and statistics, the coefficient of variation (CV) is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation <math>\ \sigma </math> to the mean <math>\ \mu </math>. It is also known as unitized risk^{[1]} or the variation coefficient. The absolute value of the CV is sometimes known as relative standard deviation (RSD), which is expressed as a percentage.
Contents
Definition
The coefficient of variation (CV) is defined as the ratio of the standard deviation <math>\ \sigma </math> to the mean <math>\ \mu </math>:
 <math>c_v = \frac{\sigma}{\mu}</math>
It shows the extent of variability in relation to the mean of the population.
The coefficient of variation should be computed only for data measured on a ratio scale, as these are the measurements that can only take nonnegative values. The coefficient of variation may not have any meaning for data on an interval scale.^{[2]} For example, most temperature scales (e.g., Celsius, Fahrenheit etc.) are interval scales that can take both positive and negative values, whereas the Kelvin temperature can never be less than zero, which is the complete absence of thermal energy. Hence, the Kelvin scale is a ratio scale. While the standard deviation (SD) can be derived on both the Kelvin and the Celsius scale (with both leading to the same SDs), the CV is only relevant as a measure of relative variability for the Kelvin scale.
Measurements that are lognormally distributed exhibit stationary CV; in contrast, SD varies depending upon the expected value of measurements.
A nonparametric possibility is the quartile coefficient of dispersion, i.e. interquartile range <math> {Q_3  Q_1} </math> divided by the median <math> {Q_2} </math>.
Estimation
When only a sample of data from a population is available, the population CV can be estimated using the ratio of the sample standard deviation <math>s \,</math> to the sample mean <math>\bar{x}</math>:
 <math>\widehat{c_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^{[3]} for a sample of size n is:
 <math>\widehat{c_v}^*=\bigg(1+\frac{1}{4n}\bigg)\widehat{c_v}</math>
Lognormal data
In many applications, it can be assumed that data are lognormally distributed (evidenced by the presence of skewness in the sampled data).^{[4]} In such cases, a more accurate estimate, derived from the properties of the lognormal distribution,^{[5]}^{[6]}^{[7]} is defined as:
 <math>\widehat{c_v}_{ln} = \sqrt{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{c_v}_{ln} \,</math> remains the same.^{[8]}) This estimate is sometimes referred to as the “geometric CV”^{[9]}^{[10]} in order to distinguish it from the simple estimate above. However, "geometric coefficient of variation" has also been defined by Kirkwood^{[11]} as:
 <math>GCV_K = {e^{s_{ln}}\!\!1}</math>
This term was intended to be analogous to the coefficient of variation, for describing multiplicative variation in lognormal data, but this definition of GCV has no theoretical basis as an estimate of <math>c_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 lognormally distributed data. If necessary, this can be derived from an estimate of <math>c_v \,</math> or GCV by inverting the corresponding formula.
Comparison to standard deviation
Advantages
The coefficient of variation is useful because the standard deviation of data must always be understood in the context of the mean of the data. In contrast, the actual value of the CV is independent of the unit in which the measurement has been taken, so it is a dimensionless number. For comparison between data sets with different units or widely different means, one should use the coefficient of variation instead of the standard deviation.
Disadvantages
 When the mean value is close to zero, the coefficient of variation will approach infinity and is therefore sensitive to small changes in the mean. This is often the case if the values do not originate from a ratio scale.
 Unlike the standard deviation, it cannot be used directly to construct confidence intervals for the mean.
 CVs are not an ideal index of the certainty of a measurement when the number of replicates varies across samples because CV is invariant to the number of replicates while certainty of the mean improves with increasing replicates. In this case standard error in percent is suggested to be superior.^{[12]}
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 lowvariance, while those with CV > 1 (such as a hyperexponential distribution) are considered highvariance. 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 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.
Laboratory measures of intraassay and interassay CVs
CV measures are often used as quality controls for quantitative laboratory assays. While intraassay and interassay 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 interassay CV estimates, it has been suggested that these practices are incorrect and that a more complex computational process is required.^{[13]} 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.^{[14]}
Distribution
Provided that negative and small positive values of the sample mean occur with negligible frequency, the probability distribution of the coefficient of variation for a sample of size n has been shown by Hendricks and Robey^{[15]} to be
 <math>dF_{c_v}=\frac{2}{\pi^{1/2}\Gamma\Big(\frac{n1}{2}\Big)}e^{\frac{n}{2(\frac{\sigma}{\mu})^2}\frac{{c_v}^2}{1+{c_v}^2}}\frac{{c_v}^{n2}}{(1+{c_v}^2)^{n/2}}\sideset{}{^\prime}\sum_{i=0}^{n1}\frac{(n1)!\Gamma\Big(\frac{ni}{2}\Big)}{(n1i)!i!}\frac{n^{i/2}}{2^{i/2}(\frac{\sigma}{\mu})^i}\frac{1}{(1+{c_v}^2)^{i/2}}dc_v ,</math>
where the symbol <math>\sideset{}{^\prime}\sum</math> indicates that the summation is over only even values of n1i, i.e., if n is odd, sum over even values of i and if n is even, sum only over odd values of i.
This is useful, for instance, in the construction of hypothesis tests or confidence intervals. Statistical inference for the coefficient of variation in normally distributed data is often based on McKay's chisquare approximation for the coefficient of variation ^{[16]}^{[17]}^{[18]}^{[19]}
Alternative
According to Liu (2012),^{[20]} Lehmann (1986)^{[21]} "also derived the sample distribution of CV in order to give an exact method for the construction of a confidence interval for CV;" it is based on a noncentral tdistribution.
Similar ratios
Standardized moments are similar ratios, <math>{\mu_k}/{\sigma^k}</math> where <math>\mu_k</math> is the k^{th} moment about the mean, which are also dimensionless and scale invariant. The variancetomean ratio, <math>\sigma^2/\mu</math>, is another similar ratio, but is not dimensionless, and hence not scale invariant. See Normalization (statistics) for further ratios.
In signal processing, particularly image processing, the reciprocal ratio <math>\mu/\sigma</math> is referred to as the signal to noise ratio.
 Relative standard deviation, <math> \sigma / \mu </math>
 Efficiency, <math>\sigma^2 / \mu^2</math>
 Standardized moment, <math>\mu_k/\sigma^k</math>
 Variancetomean ratio, <math>\sigma^2/\mu</math>
 Fano factor, <math>\sigma^2_W/\mu_W</math> (windowed VMR)
 Signaltonoise ratio, <math>\mu/\sigma</math> (in signal processing)
See also
References
 ^ Broverman, Samuel A. (2001). Actex study manual, Course 1, Examination of the Society of Actuaries, Exam 1 of the Casualty Actuarial Society (2001 ed. ed.). Winsted, CT: Actex Publications. p. 104. ISBN 9781566983969. Retrieved 7 June 2014.
 ^ "What is the difference between ordinal, interval and ratio variables? Why should I care?". GraphPad Software Inc. Retrieved 20080222.
 ^ Sokal RR & Rohlf FJ. Biometry (3rd Ed). New York: Freeman, 1995. p. 58. ISBN 0716724111
 ^ Limpert, Eckhard; Stahel, Werner A.; Abbt, Markus (2001). "Lognormal Distributions across the Sciences: Keys and Clues". BioScience 51: 341–352. doi:10.1641/00063568(2001)051[0341:LNDATS]2.0.CO;2.
 ^ Koopmans, L. H.; Owen, D. B.; Rosenblatt, J. I. (1964). "Confidence intervals for the coefficient of variation for the normal and log normal distributions". Biometrika 51: 25. doi:10.1093/biomet/51.12.25.
 ^ Diletti, E; Hauschke, D; Steinijans, VW (1992). "Sample size determination for bioequivalence assessment by means of confidence intervals". International journal of clinical pharmacology, therapy, and toxicology. 30 Suppl 1: S51–8. PMID 1601532.
 ^ Julious, Steven A.; Debarnot, Camille A. M. (2000). "Why Are Pharmacokinetic Data Summarized by Arithmetic Means?". Journal of Biopharmaceutical Statistics 10 (1): 55–71. PMID 10709801. doi:10.1081/BIP100101013.
 ^ Reed JF, Lynn F, Meade BD. (2002) "Use of Coefficient of Variation in Assessing Variability of Quantitative Assays". Clin Diagn Lab Immunol. 9(6): 1235–1239. doi:10.1128/CDLI.9.6.12351239.2002
 ^ Sawant,S.; Mohan, N. (2011) "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS", PharmaSUG2011, Paper PO08
 ^ Schiff MH, et al. Ann Rheum Dis 2014;0:1–3. doi:10.1136/annrheumdis2014205228
 ^ Kirkwood, TBL (1979). "Geometric means and measures of dispersion". Biometrics 35: 908–9. doi:10.2307/2530139.
 ^ Eisenberg, Dan (2015). "Improving qPCR telomere length assays: Controlling for well position effects increases statistical power". American Journal of Human Biology. PMID 25757675. doi:10.1002/ajhb.22690.
 ^ Rodbard, D (October 1974). "Statistical quality control and routine data processing for radioimmunoassays and immunoradiometric assays.". Clinical chemistry 20 (10): 1255–70. PMID 4370388.
 ^ Eisenberg, Dan (2015). "Improving qPCR telomere length assays: Controlling for well position effects increases statistical power". American Journal of Human Biology. PMID 25757675. doi:10.1002/ajhb.22690.
 ^ Hendricks, Walter A.; Robey, Kate W. (1936). "The Sampling Distribution of the Coefficient of Variation". The Annals of Mathematical Statistics 7 (3): 129–32. JSTOR 2957564. doi:10.1214/aoms/1177732503.
 ^ Iglevicz, Boris; Myers, Raymond (1970). "Comparisons of approximations to the percentage points of the sample coefficient of variation". Technometrics 12 (1): 166–169. JSTOR 1267363.
 ^ Bennett, B. M. (1976). On an approximate test for homogeneity of coefficients of variation. In: Ziegler, W. J. (ed.) Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl. 22, 169171
 ^ Vangel, Mark G. (1996). "Confidence intervals for a normal coefficient of variation". The American Statistician 50 (1): 21–26. JSTOR 2685039. doi:10.1080/00031305.1996.10473537..
 ^ Forkman, Johannes. "Estimator and tests for common coefficients of variation in normal distributions" (PDF). Communications in Statistics  Theory and Methods 38. pp. 21–26. Retrieved 20130923.
 ^ [1], p.3
 ^ Lehmann, E. L. (1986), Testing Statistical Hypothesis. 2nd ed. New York: Wiley.
