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 to the mean (or its absolute value, ). 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]} In addition, CV is utilized by economists and investors in economic models and in determining the volatility of a security^{[citation needed]}.
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✪ Coefficient of variation

✪ What is the Coefficient Of Variation?? (+ examples!)

✪ Coefficient of variation (part 1)

✪ Standard deviation and Coefficient of variation

✪ Coefficient of Variation Example and Explanation
Transcription
This is Stephanie from StatisticsHowTo.com and in this video I'll be showing you how to find the coefficient of variation. The coefficient of variation is used when you want to compare scores from data sets with significantly different means. Let me show you an example. Let's say, your State Standardized Test, let's just call it the "ST" for the standardized test. You know the mean for the standardized test is 71. If school one scores, say the sample mean here. Let say the average score, at school one was 62. And the average score at school two, that's my sample mean, say the average score was 81. Then you probably conclude that school two is better than school one. This could this be from many reasons. It could be the difference in the quality of teachers. It could be a lack of funding to certain schools. A different curriculum, it could be many reasons. But you can compare these scores directly. And say that school two is definitely better than school one as far as the standardized test goes. What if score one wasn't taking standardized test? What if it was taking a slightly modified version of the test? That's when you use the coefficient of variation to compare the two scores. So let's say we have two versions of the test. Let's say the one version of the test, let's just call this the Regular test has preset answers. And a second version of the test has randomized answers. The means for these two tests are completely different. We're gonna write the mean as u. And the regular test my mean is 50.1. And for the randomized answers the mean is 45.8. I know that my standard deviation for these tests is 11.2 and 12.9. And just looking at this set of data you can't directly compare any particular score that a student gets on this test. Because the population means are completely different and the standard deviations are different as well. This is when you use the coefficient of variation. So how do we use this formula? You divide standard deviation by the mean and multiply it by 100% using this top formula here. For the regular test, I have standard deviation of 11.2 divided by my mean of 50.1, gonna multiply that by 100% that gives me 22.355%. For the second test, my random, I've got 12.9 divided by 45.8 that's my mean. Multiply by 100%, gives me 28.266%. Now I can directly compare the two tests with these percentages. The coefficient of variation formula for a sample works exactly the same way. The "s" is the standard deviation for the sample. And "xbar" is the mean for the sample. Visit us at StatisticsHowTo.com for more videos and articles on Elementary Statistics and AP Statistics.
Contents
Definition
The coefficient of variation (CV) is defined as the ratio of the standard deviation to the mean :^{[1]} 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 allow the division operation. 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 with arbitrary zeros, so the coefficient of variation would be different depending on which scale you used. On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale. While the standard deviation (SD) can be meaningfully derived using Kelvin, Celsius, or Fahrenheit, the CV is only valid as a measure of relative variability for the Kelvin scale because its computation involves division.
Measurements that are lognormally 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, half the interquartile range divided by the average of the quartiles (the midhinge), .
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 nonlinear and include a continuous range for the independent variable with sparse measurements across each value (e.g., scatterplot) may be amenable to single CV calculation using a maximumlikelihood estimation approach.^{[3]}
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 = 10%
A data set of [1, 5, 6, 8, 10, 40, 65, 88] has still more variability. Its standard deviation is 32.9 and its average is 27.9, giving a coefficient of variation of
 32.9 / 27.9 = 118%
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 = 79%. For the second set (which are the same temperatures) it is 28.46/68 = 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 5.39%.
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 to the sample mean :
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^{[4]} for a sample of size n is:
Lognormal data
In many applications, it can be assumed that data are lognormally distributed (evidenced by the presence of skewness in the sampled data).^{[5]} In such cases, a more accurate estimate, derived from the properties of the lognormal distribution,^{[6]}^{[7]}^{[8]} is defined as:
where 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 is converted to base e using , and the formula for remains the same.^{[9]}) This estimate is sometimes referred to as the "geometric CV"^{[10]}^{[11]} in order to distinguish it from the simple estimate above. However, "geometric coefficient of variation" has also been defined by Kirkwood^{[12]} as:
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 itself.
For many practical purposes (such as sample size determination and calculation of confidence intervals) it is which is of most use in the context of lognormally distributed data. If necessary, this can be derived from an estimate of 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 measurement when the number of replicates varies across samples because CV is invariant to the number of replicates while the certainty of the mean improves with increasing replicates. In this case, standard error in percent is suggested to be superior.^{[13]}
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^{[citation needed]}. 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.^{[14]}
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.^{[15]}
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.^{[16]} 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.^{[13]} 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.^{[17]}
As a measure of economic inequality
The coefficient of variation fulfills the requirements for a measure of economic inequality.^{[18]}^{[19]}^{[20]} If x (with entries x_{i}) is a list of the values of an economic indicator (e.g. wealth), with x_{i} being the wealth of agent i, then the following requirements are met:
 Anonymity – c_{v} 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_{v}(x)=c_{v}(αx) where α is a real number.^{[20]}
 Population independence – If {x,x} is the list x appended to itself, then c_{v}({x,x})=c_{v}(x). This follows from the fact that the variance and mean both obey this principle.
 PigouDalton transfer principle: when wealth is transferred from a wealthier agent i to a poorer agent j (i.e. x_{i} > x_{j}) without altering their rank, then c_{v} decreases and vice versa.^{[20]}
c_{v} assumes its minimum value of zero for complete equality (all x_{i} are equal).^{[20]} 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).^{[20]} 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^{[21]} to be
where the symbol 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 ^{[22]}^{[23]}^{[24]}^{[25]}^{[26]}^{[27]}
Alternative
According to Liu (2012),^{[28]} Lehmann (1986).^{[29]} "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, where is the k^{th} moment about the mean, which are also dimensionless and scale invariant. The variancetomean ratio, , 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 is referred to as the signal to noise ratio in general and signaltonoise ratio (imaging) in particular.
 Efficiency,
 Standardized moment,
 Variancetomean ratio (or relative variance),
 Fano factor, (windowed VMR)
 Relative Standard Error
See also
References
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External links
 cvequality: R package to test for significant differences between multiple coefficients of variation