In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), percent RMS, and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution.
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,[citation needed] by economists and investors in economic models, and in psychology/neuroscience.
The coefficient of variation (CV) is defined as the ratio of the standard deviation
On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale.
While a standard deviation (SD) can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale.
A more robust possibility is the quartile coefficient of dispersion, half the interquartile range
In most cases, a CV is computed for a single independent variable (e.g., a single factory product) with numerous, repeated measures of a dependent variable (e.g., error in the production process).
However, data that are linear or even logarithmically non-linear and include a continuous range for the independent variable with sparse measurements across each value (e.g., scatter-plot) may be amenable to single CV calculation using a maximum-likelihood estimation approach.
[5] In such cases, a more accurate estimate, derived from the properties of the log-normal distribution,[6][7][8] is defined as: where
is the sample standard deviation of the data after a natural log transformation.
For many practical purposes (such as sample size determination and calculation of confidence intervals) it is
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.
Some formulas in these fields are expressed using the squared coefficient of variation, often abbreviated SCV.
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.
[13] 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.
The term is used widely in the design of pollution control equipment, such as electrostatic precipitators (ESPs),[15] selective catalytic reduction (SCR), scrubbers, and similar devices.
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.
[17] 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.
[19] The coefficient of variation fulfills the requirements for a measure of economic inequality.
[22] 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).
Archaeologists often use CV values to compare the degree of standardisation of ancient artefacts.
[23][24] Variation in CVs has been interpreted to indicate different cultural transmission contexts for the adoption of new technologies.
[25] Coefficients of variation have also been used to investigate pottery standardisation relating to changes in social organisation.
[26] Archaeologists also use several methods for comparing CV values, for example the modified signed-likelihood ratio (MSLR) test for equality of CVs.
[27][28] 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.
[30][31][32][33][34][35] Liu (2012) reviews methods for the construction of a confidence interval for the coefficient of variation.
[36] Notably, Lehmann (1986) derived the sampling distribution for the coefficient of variation using a non-central t-distribution to give an exact method for the construction of the CI.