Coefficient of variation

In probability theory and statistics, the coefficient of variation, also known as normalized root-mean-square deviation, percent RMS, and relative standard deviation, is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation to the mean . 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, by economists and investors in economic models, in epidemiology, and in psychology/neuroscience.

Definition

The coefficient of variation is defined as the ratio of the standard deviation to the mean,
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 scales that have a meaningful zero and hence allow relative comparison of two measurements. The coefficient of variation may not have any meaning for data on an interval scale. For example, most temperature scales are interval scales with arbitrary zeros, so the computed coefficient of variation would be different depending on the scale used. On the other hand, Kelvin temperature has a meaningful zero, the complete absence of thermal energy, and thus is a ratio scale. In plain language, it is meaningful to say that 20 Kelvin is twice as hot as 10 Kelvin, but only in this scale with a true absolute zero. While a standard deviation can be measured in Kelvin, Celsius, or Fahrenheit, the value computed is only applicable to that scale. Only the Kelvin scale can be used to compute a valid coefficient of variation.
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, half the interquartile range divided by the average of the quartiles,.
In most cases, a CV is computed for a single independent variable with numerous, repeated measures of a dependent variable. 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 may be amenable to single CV calculation using a maximum-likelihood estimation approach.

Examples

In the examples below, we will take the values given as randomly chosen from a larger population of values.

The data set has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0
The data set has more variability. Its standard deviation is 10 and its average is 100, giving the coefficient of variation as 10 / 100 = 0.1
The data set 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 = 1.18

In these examples, we will take the values given as the entire population of values.

The data set has a population standard deviation of 0 and a coefficient of variation of 0 / 100 = 0
The data set has a population standard deviation of 8.16 and a coefficient of variation of 8.16 / 100 = 0.0816
The data set has a population standard deviation of 30.8 and a coefficient of variation of 30.8 / 27.9 = 1.10
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 for a sample of size n is:

Log-normal data

Many datasets follow an approximately log-normal distribution. In such cases, a more accurate estimate, derived from the properties of the log-normal distribution, is defined as:
where is the sample standard deviation of the data after a natural log transformation. 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:
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 itself.
For many practical purposes it is which is of most use in the context of log-normally 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.
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 are considered low-variance, while those with CV > 1 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. Essentially the CV replaces the standard deviation term with the Root Mean Square Deviation. 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.
In fluid dynamics, the CV, also referred to as Percent RMS, %RMS, %RMS Uniformity, or Velocity RMS, is a useful determination of flow uniformity for industrial processes. The term is used widely in the design of pollution control equipment, such as electrostatic precipitators, selective catalytic reduction, scrubbers, and similar devices. The Institute of Clean Air Companies references RMS deviation of velocity in the design of fabric filters. The guiding principle is that many of these pollution control devices require "uniform flow" entering and through the control zone. This can be related to uniformity of velocity profile, temperature distribution, gas species, and other flow-related parameters. The Percent RMS also is used to assess flow uniformity in combustion systems, HVAC systems, ductwork, inlets to fans and filters, air handling units, etc. where performance of the equipment is influenced by the incoming flow distribution.

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 is a list of the values of an economic indicator, 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 = c_v where α is a real number.
Population independence – If is the list x appended to itself, then c_v = c_v. 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 without altering their rank, then c_v decreases and vice versa.

c_v assumes its minimum value of zero for complete equality. Its most notable drawback is that it is not bounded from above, so it cannot be normalized to be within a fixed range. It is, however, more mathematically tractable than the Gini coefficient.