Coefficient of variation - wikipedia Coefficient of variation Not to be confused with Coefficient of determination. In probability theory and statistics, the coefficient of variation ( CV ), also known as relative standard deviation ( RSD ), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is often expressed as a percentage, and is defined as the ratio of the standard deviation σ ( displaystyle sigma ) to the mean μ ( displaystyle mu ) ( or its absolute value, μ ( displaystyle mu ) ). The CV or RSD is widely used in analytical chemistry to express the precision and repeatability of an assay. It is also commonly used in fields such as engineering or physics when doing quality assurance studies and ANOVA gauge R&R. In addition, CV is utilized by economists and investors in economic models and in determining the volatility of a security. Contents 1 Definition 2 Examples 3 Examples of misuse 4 Estimation 4.1 Log - normal data 5 Comparison to standard deviation 5.1 Advantages 5.2 Disadvantages 6 Applications 6.1 Laboratory measures of intra-assay and inter-assay CVs 6.2 As a measure of economic inequality 7 Distribution 7.1 Alternative 8 Similar ratios 9 See also 10 References Definition The coefficient of variation ( CV ) is defined as the ratio of the standard deviation σ ( displaystyle sigma ) to the mean μ ( displaystyle mu ) : : c v = σ μ ( displaystyle c_ ( \rm ( v ) ) = ( \frac ( sigma ) ( mu ) ) ) 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 non-negative values. The coefficient of variation may not have any meaning for data on an interval scale. 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 log - normally distributed exhibit stationary CV ; in contrast, SD varies depending upon the expected value of measurements. A more robust possibility is the quartile coefficient of dispersion, i.e. half the interquartile range ( Q 3 − Q 1 ) / 2 ( displaystyle ( ( Q_ ( 3 ) - Q_ ( 1 ) ) / 2 ) ) divided by the average of the quartiles ( the midhinge ), ( Q 1 + Q 3 ) / 2 ( displaystyle ( ( Q_ ( 1 ) + Q_ ( 3 ) ) / 2 ) ). Examples A data set of ( 100, 100, 100 ) has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0. A data set of ( 90, 100, 110 ) has more variability. Its standard deviation is 10 and its average is 100, giving the coefficient of variation as 10 / 100 = 0.1 A data set of ( 1, 5, 6, 8, 10, 40, 65, 88 ) has still more variability. Its standard deviation is 32.904 and its average is 27.875, giving a coefficient of variation of 32.904 / 27.875 = 1.180. Examples of misuse Comparing coefficients of variation between parameters using relative units can result in differences that may not be real. If we compare the same set of temperatures in Celsius and Fahrenheit ( both relative units, where kelvin and Rankine scale are their associated absolute values ) : Celsius : ( 0, 10, 20, 30, 40 ) Fahrenheit : ( 32, 50, 68, 86, 104 ) The sample standard deviations are 15.81 and 28.46, respectively. The CV of the first set is 15.81 / 20 = 0.79. For the second set ( which are the same temperatures ) it is 28.46 / 68 = 0.42. If, for example, the data sets are temperature readings from two different sensors ( a Celsius sensor and a Fahrenheit sensor ) and you want to know which sensor is better by picking the one with the least variance, then you will be misled if you use CV. The problem here is that you have divided by a relative value rather than an absolute. Comparing the same data set, now in absolute units : Kelvin : ( 273.15, 283.15, 293.15, 303.15, 313.15 ) Rankine : ( 491.67, 509.67, 527.67, 545.67, 563.67 ) The sample standard deviations are still 15.81 and 28.46, respectively, because the standard deviation is not affected by a constant offset. The coefficients of variation, however, are now both equal to 0.0539. 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 s ( displaystyle s , ) to the sample mean x  ̄ ( displaystyle ( \bar ( x ) ) ) : c v ^ = s x  ̄ ( displaystyle ( widehat ( c_ ( \rm ( v ) ) ) ) = ( \frac ( s ) ( \bar ( x ) ) ) ) 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 for a sample of size n is : c v ^ ∗ = ( 1 + 1 4 n ) c v ^ ( displaystyle ( widehat ( c_ ( \rm ( v ) ) ) ) ^ ( * ) = ( \bigg ( ) 1 + ( \frac ( 1 ) ( 4n ) ) ( \bigg ) ) ( widehat ( c_ ( \rm ( v ) ) ) ) ) Log - normal data In many applications, it can be assumed that data are log - normally distributed ( evidenced by the presence of skewness in the sampled data ). In such cases, a more accurate estimate, derived from the properties of the log - normal distribution, is defined as : c v ^ l n = e s l n 2 − 1 ( displaystyle ( widehat ( c_ ( \rm ( v ) ) ) ) _ ( \rm ( ln ) ) = ( sqrt ( mathrm ( e ) ^ ( s_ ( \rm ( ln ) ) ^ ( 2 ) ) - 1 ) ) ) where s l n ( displaystyle ( s_ ( \rm ( ln ) ) ) , ) 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 s b ( displaystyle s_ ( b ) , ) is converted to base e using s l n = s b ln ⁡ ( b ) ( displaystyle s_ ( \rm ( ln ) ) = s_ ( b ) ln ( b ) , ), and the formula for c v ^ l n ( displaystyle ( widehat ( c_ ( \rm ( v ) ) ) ) _ ( \rm ( ln ) ) , ) remains the same. ) This estimate is sometimes referred to as the `` geometric CV '' in order to distinguish it from the simple estimate above. However, `` geometric coefficient of variation '' has also been defined by Kirkwood as : G C V K = e s l n − 1 ( displaystyle mathrm ( GCV_ ( K ) ) = ( mathrm ( e ) ^ ( s_ ( \rm ( ln ) ) ) ! ! - 1 ) ) 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 c v ( displaystyle c_ ( \rm ( v ) ) , ) itself. For many practical purposes ( such as sample size determination and calculation of confidence intervals ) it is s l n ( displaystyle s_ ( ln ) , ) which is of most use in the context of log - normally distributed data. If necessary, this can be derived from an estimate of c v ( displaystyle c_ ( \rm ( v ) ) , ) 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 can not 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. 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. 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. In actuarial science, the CV is known as unitized risk. 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. Laboratory measures of intra-assay and inter-assay CVs CV measures are often used as quality controls for quantitative laboratory assays. 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. 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. 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. As a measure of economic Inequality The coefficient of variation fulfills the requirements for a measure of economic inequality. If x ( with entries x ) is a list of the values of an economic indicator ( e.g. wealth ), with x being the wealth of agent i, then the following requirements are met : Anonymity -- c 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 ( x ) = c ( αx ) where α is a real number. Population independence -- If ( x, x ) is the list x appended to itself, then c ( ( x, x ) ) = c ( 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 > x ) without altering their rank, then c decreases and vice versa. c assumes its minimum value of zero for complete equality ( all x are equal ). Its most notable drawback is that it is not bounded from above, so it can not be normalized to be within a fixed range ( e.g. like the Gini coefficient which is constrained to be between 0 and 1 ). It is, however, more mathematically tractable than the Gini Coefficient. 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 to be d F c v = 2 π 1 / 2 Γ ( n − 1 2 ) e − n 2 ( σ μ ) 2 c v 2 1 + c v 2 c v n − 2 ( 1 + c v 2 ) n / 2 ∑ ∑ ′ i = 0 n − 1 ⁡ ( n − 1 ) ! Γ ( n − i 2 ) ( n − 1 − i ) ! i ! n i / 2 2 i / 2 ( σ μ ) i 1 ( 1 + c v 2 ) i / 2 d c v, ( displaystyle mathrm ( d ) F_ ( c_ ( \rm ( v ) ) ) = ( \frac ( 2 ) ( pi ^ ( 1 / 2 ) Gamma left ( ( \frac ( n - 1 ) ( 2 ) ) \right ) ) ) ; mathrm ( e ) ^ ( - ( \frac ( n ) ( 2 left ( ( \frac ( sigma ) ( mu ) ) \right ) ^ ( 2 ) ) ) ( \frac ( ( c_ ( \rm ( v ) ) ) ^ ( 2 ) ) ( 1 + ( c_ ( \rm ( v ) ) ) ^ ( 2 ) ) ) ) ( \frac ( ( c_ ( \rm ( v ) ) ) ^ ( n - 2 ) ) ( ( 1 + ( c_ ( \rm ( v ) ) ) ^ ( 2 ) ) ^ ( n / 2 ) ) ) sideset ( ) ( ^ ( prime ) ) sum _ ( i = 0 ) ^ ( n - 1 ) ( \frac ( ( n - 1 ) ! , Gamma left ( ( \frac ( n-i ) ( 2 ) ) \right ) ) ( ( n - 1 - i ) ! , i ! , ) ) ( \frac ( n ^ ( i / 2 ) ) ( 2 ^ ( i / 2 ) left ( ( \frac ( sigma ) ( mu ) ) \right ) ^ ( i ) ) ) ( \frac ( 1 ) ( ( 1 + ( c_ ( \rm ( v ) ) ) ^ ( 2 ) ) ^ ( i / 2 ) ) ) , mathrm ( d ) c_ ( \rm ( v ) ), ) where the symbol ∑ ∑ ′ ( displaystyle sideset ( ) ( ^ ( prime ) ) sum ) indicates that the summation is over only even values of n - 1 - i, 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 chi - square approximation for the coefficient of variation Alternative According to Liu ( 2012 ), Lehmann ( 1986 ). `` 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 non-central t - distribution. Similar ratios Standardized moments are similar ratios, μ k / σ k ( displaystyle ( mu _ ( k ) ) / ( sigma ^ ( k ) ) ) where μ k ( displaystyle mu _ ( k ) ) is the k moment about the mean, which are also dimensionless and scale invariant. The variance - to - mean ratio, σ 2 / μ ( displaystyle sigma ^ ( 2 ) / mu ), 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 μ / σ ( displaystyle mu / sigma ) is referred to as the signal to noise ratio in general and signal - to - noise ratio ( imaging ) in particular. Efficiency, σ 2 / μ 2 ( displaystyle sigma ^ ( 2 ) / mu ^ ( 2 ) ) Standardized moment, μ k / σ k ( displaystyle mu _ ( k ) / sigma ^ ( k ) ) Variance - to - mean ratio ( or relative variance ), σ 2 / μ ( displaystyle sigma ^ ( 2 ) / mu ) Fano factor, σ W 2 / μ W ( displaystyle sigma _ ( W ) ^ ( 2 ) / mu _ ( W ) ) ( windowed VMR ) Relative Standard Error See also Omega ratio Sampling ( statistics ) References Jump up ^ Everitt, Brian ( 1998 ). The Cambridge Dictionary of Statistics. Cambridge, UK New York : Cambridge University Press. ISBN 0521593468. Jump up ^ `` What is the difference between ordinal, interval and ratio variables ? Why should I care ? ''. GraphPad Software Inc. Retrieved 22 February 2008. Jump up ^ Sokal RR & Rohlf FJ. Biometry ( 3rd Ed ). New York : Freeman, 1995. p. 58. ISBN 0 - 7167 - 2411 - 1 Jump up ^ Limpert, Eckhard ; Stahel, Werner A. ; Abbt, Markus ( 2001 ). `` Log - normal Distributions across the Sciences : Keys and Clues ''. BioScience. 51 : 341 -- 352. doi : 10.1641 / 0006 - 3568 ( 2001 ) 051 ( 0341 : LNDATS ) 2.0.CO ; 2. Jump up ^ 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 -- 32. doi : 10.1093 / biomet / 51.1 - 2.25. Jump up ^ 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. Jump up ^ Julious, Steven A. ; Debarnot, Camille A.M. ( 2000 ). `` Why Are Pharmacokinetic Data Summarized by Arithmetic Means ? ''. Journal of Biopharmaceutical Statistics. 10 ( 1 ) : 55 -- 71. doi : 10.1081 / BIP - 100101013. PMID 10709801. Jump up ^ 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. 1235 - 1239.2002. Jump up ^ Sawant, S. ; Mohan, N. ( 2011 ) `` FAQ : Issues with Efficacy Analysis of Clinical Trial Data Using SAS '', PharmaSUG2011, Paper PO08 Jump up ^ Schiff, MH ; et al. ( 2014 ). `` 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 ''. Ann Rheum Dis. 73 : 1 -- 3. doi : 10.1136 / annrheumdis - 2014 - 205228. Jump up ^ Kirkwood, TBL ( 1979 ). `` Geometric means and measures of dispersion ''. Biometrics. 35 : 908 -- 9. 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( 30 August 2016 ). `` Telomere length measurement validity : the coefficient of variation is invalid and can not be used to compare quantitative polymerase chain reaction and Southern blot telomere length measurement technique ''. International Journal of Epidemiology : dyw191. doi : 10.1093 / ije / dyw191. ISSN 0300 - 5771. Jump up ^ Champernowne, D.G. ; Cowell, F.A. ( 1999 ). Economic Inequality and Income Distribution. Cambridge University Press. Jump up ^ Campano, F. ; Salvatore, D. ( 2006 ). Income distribution. Oxford University Press. ^ Jump up to : Bellu, Lorenzo Giovanni ; Liberati, Paolo ( 2006 ). `` Policy Impacts on Inequality -- Simple Inequality Measures '' ( PDF ). EASYPol, Analytical tools. Policy Support Service, Policy Assistance Division, FAO. Retrieved 13 June 2016. Jump up ^ Hendricks, Walter A. ; Robey, Kate W. ( 1936 ). `` The Sampling Distribution of the Coefficient of Variation ''. The Annals of Mathematical Statistics. 7 ( 3 ) : 129 -- 32. doi : 10.1214 / aoms / 1177732503. JSTOR 2957564. Jump up ^ Iglevicz, Boris ; Myers, Raymond ( 1970 ). `` Comparisons of approximations to the percentage points of the sample coefficient of variation ''. Technometrics. 12 ( 1 ) : 166 -- 169. doi : 10.2307 / 1267363. JSTOR 1267363. Jump up ^ Bennett, B.M. ( 1976 ). `` On an approximate test for homogeneity of coefficients of variation ''. Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl. 22 : 169 -- 171. Jump up ^ Vangel, Mark G. ( 1996 ). `` Confidence intervals for a normal coefficient of variation ''. The American Statistician. 50 ( 1 ) : 21 -- 26. doi : 10.1080 / 00031305.1996. 10473537. JSTOR 2685039... Jump up ^ Forkman, Johannes. `` Estimator and tests for common coefficients of variation in normal distributions '' ( PDF ). Communications in Statistics -- Theory and Methods. pp. 21 -- 26. doi : 10.1080 / 03610920802187448. 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Statistics Outline Index Descriptive statistics Continuous data Center Mean arithmetic geometric harmonic Median Mode Dispersion Variance Standard deviation Coefficient of variation Percentile Range Interquartile range Shape Central limit theorem Moments Skewness Kurtosis L - moments Count data Index of dispersion Summary tables Grouped data Frequency distribution Contingency table Dependence Pearson product - moment correlation Rank correlation Spearman 's rho Kendall 's tau Partial correlation Scatter plot Graphics Bar chart Biplot Box plot Control chart Correlogram Fan chart Forest plot Histogram Pie chart Q -- Q plot Run chart Scatter plot Stem - and - leaf display Radar chart Data collection Study design Population Statistic Effect size Statistical power Sample size determination Missing data Survey methodology Sampling stratified cluster Standard error Opinion poll Questionnaire Controlled experiments Design control optimal Controlled trial Randomized Random assignment Replication Blocking Interaction Factorial experiment Uncontrolled studies Observational study Natural experiment Quasi-experiment Statistical inference Statistical theory Population Statistic Probability distribution Sampling distribution Order statistic Empirical distribution Density estimation Statistical model L space Parameter location scale shape Parametric family Likelihood ( monotone ) Location -- scale family Exponential family Completeness Sufficiency Statistical functional Bootstrap U V Optimal decision loss function Efficiency Statistical distance divergence Asymptotics Robustness Frequentist inference Point estimation Estimating equations Maximum likelihood Method of moments M - estimator Minimum distance Unbiased estimators Mean - unbiased minimum - variance Rao -- Blackwellization Lehmann -- Scheffé theorem Median unbiased Plug - in Interval estimation Confidence interval Pivot Likelihood interval Prediction interval Tolerance interval Resampling Bootstrap Jackknife Testing hypotheses 1 - & 2 - tails Power Uniformly most powerful test Permutation test Randomization test Multiple comparisons Parametric tests Likelihood - ratio Wald Score Specific tests Z ( normal ) Student 's t - test Goodness of fit Chi - squared Kolmogorov -- Smirnov Anderson -- Darling Lilliefors Jarque -- Bera Normality ( Shapiro -- Wilk ) Likelihood - ratio test Model selection Cross validation AIC BIC Rank statistics Sign Sample median Signed rank ( Wilcoxon ) Hodges -- Lehmann estimator Rank sum ( Mann -- Whitney ) Nonparametric anova 1 - way ( Kruskal -- Wallis ) 2 - way ( Friedman ) Ordered alternative ( Jonckheere -- Terpstra ) Bayesian inference Bayesian probability prior posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator Correlation Regression analysis Correlation Pearson product - moment Partial correlation Confounding variable Coefficient of determination Regression analysis Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines ( MARS ) Linear regression Simple linear regression Ordinary least squares General linear model Bayesian regression Non-standard predictors Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity Generalized linear model Exponential families Logistic ( Bernoulli ) / Binomial / Poisson regressions Partition of variance Analysis of variance ( ANOVA, anova ) Analysis of covariance Multivariate ANOVA Degrees of freedom Categorical / Multivariate / Time - series / Survival analysis Categorical Cohen 's kappa Contingency table Graphical model Log - linear model McNemar 's test Multivariate Regression Manova Principal components Canonical correlation Discriminant analysis Cluster analysis Classification Structural equation model Factor analysis Multivariate distributions Elliptical distributions Normal Time - 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Question: what does the coefficient of variation tell us
Answer: dispersion of a probability distribution or frequency distribution