In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, the coefficient of variation (CV), also known as relative standard deviation (RSD), is a
standardized
Standardization or standardisation is the process of implementing and developing technical standards based on the consensus of different parties that include firms, users, interest groups, standards organizations and governments. Standardization ...
measure of
dispersion
Dispersion may refer to:
Economics and finance
*Dispersion (finance), a measure for the statistical distribution of portfolio returns
*Price dispersion, a variation in prices across sellers of the same item
*Wage dispersion, the amount of variatio ...
of a
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
or
frequency distribution
In statistics, the frequency (or absolute frequency) of an event i is the number n_i of times the observation has occurred/recorded in an experiment or study. These frequencies are often depicted graphically or in tabular form.
Types
The cumula ...
. It is often expressed as a percentage, and is defined as the ratio of the
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
to the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
(or its
absolute value
In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
, The CV or RSD is widely used in
analytical chemistry
Analytical chemistry studies and uses instruments and methods to separate, identify, and quantify matter. In practice, separation, identification or quantification may constitute the entire analysis or be combined with another method. Separati ...
to express the precision and repeatability of an
assay
An assay is an investigative (analytic) procedure in laboratory medicine, mining, pharmacology, environmental biology and molecular biology for qualitatively assessing or quantitatively measuring the presence, amount, or functional activity of a ...
. It is also commonly used in fields such as
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
or
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
when doing quality assurance studies and
ANOVA gauge R&R, by economists and investors in
economic model
In economics, a model is a theoretical construct representing economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified, often mathematical, framework desig ...
s, and in
neuroscience
Neuroscience is the scientific study of the nervous system (the brain, spinal cord, and peripheral nervous system), its functions and disorders. It is a multidisciplinary science that combines physiology, anatomy, molecular biology, development ...
.
Definition
The coefficient of variation (CV) 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 (
ratio scale
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used f ...
) and hence allow relative comparison of two measurements (i.e., division of one measurement by the other). The coefficient of variation may not have any meaning for data on an
interval scale
Level of measurement or scale of measure is a classification that describes the nature of information within the values assigned to variables. Psychologist Stanley Smith Stevens developed the best-known classification with four levels, or scal ...
. For example, most temperature scales (e.g., Celsius, Fahrenheit etc.) 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
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
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 (SD) 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 variability.
Measurements that are
log-normal
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
ly 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 In statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and is used to make comparisons within and between data sets. Since it is based on quantile information, it is less sensitive to outliers tha ...
, half the
interquartile range
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The IQR may also be called the midspread, middle 50%, fourth spread, or H‑spread. It is defined as the difference ...
divided by the average of the quartiles (the
midhinge In statistics, the midhinge is the average of the first and third quartiles and is thus a measure of location.
Equivalently, it is the 25% trimmed mid-range or 25% midsummary; it is an L-estimator.
: \operatorname(X) = \overline = \frac = \frac ...
),
.
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
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
approach.
Examples
In the examples below, we will take the values given as randomly chosen from a larger population of values.
* The data set
00, 100, 100has constant values. Its
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0
* The data set
0, 100, 110has 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
, 5, 6, 8, 10, 40, 65, 88has 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
00, 100, 100has a
population standard deviation of 0 and a coefficient of variation of 0 / 100 = 0
* The data set
0, 100, 110has a population standard deviation of 8.16 and a coefficient of variation of 8.16 / 100 = 0.0816
* The data set
, 5, 6, 8, 10, 40, 65, 88has 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
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while ...
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
In statistics, the bias of an estimator (or bias function) is the difference between this estimator's expected value and the true value of the parameter being estimated. An estimator or decision rule with zero bias is called ''unbiased''. In st ...
. For
normally distributed data, an unbiased estimator for a sample of size n is:
:
Log-normal data
In many applications, it can be assumed that data are log-normally distributed (evidenced by the presence of
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal d ...
in the sampled data). In such cases, a more accurate estimate, derived from the properties of the
log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
, is defined as:
:
where
is the sample standard deviation of the data after a
natural log
The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
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.) This estimate is sometimes referred to as the "geometric CV" (GCV) 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 (such as
sample size determination
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
and calculation of
confidence intervals
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
) 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
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
.
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 interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
s 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.
Applications
The coefficient of variation is also common in applied probability fields such as
renewal theory
Renewal theory is the branch of probability theory that generalizes the Poisson process for arbitrary holding times. Instead of exponentially distributed holding times, a renewal process may have any independent and identically distributed (IID) ho ...
,
queueing theory
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
, and
reliability theory
Reliability engineering is a sub-discipline of systems engineering that emphasizes the ability of equipment to function without failure. Reliability describes the ability of a system or component to function under stated conditions for a specifi ...
. In these fields, the
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
is often more important than the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
.
The standard deviation of an
exponential distribution
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
is equal to its mean, so its coefficient of variation is equal to 1. Distributions with CV < 1 (such as an
Erlang distribution
The Erlang distribution is a two-parameter family of continuous probability distributions with support x \in independent exponential distribution">exponential variables with mean 1/\lambda each. Equivalently, it is the distribution of the tim ...
) are considered low-variance, while those with CV > 1 (such as a
hyper-exponential distribution
In probability theory, a hyperexponential distribution is a continuous probability distribution whose probability density function of the random variable ''X'' is given by
: f_X(x) = \sum_^n f_(x)\;p_i,
where each ''Y'i'' is an exponentially ...
) 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
The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...
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
assay
An assay is an investigative (analytic) procedure in laboratory medicine, mining, pharmacology, environmental biology and molecular biology for qualitatively assessing or quantitatively measuring the presence, amount, or functional activity of a ...
s. 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
In statistics, the intraclass correlation, or the intraclass correlation coefficient (ICC), is a descriptive statistic that can be used when quantitative measurements are made on units that are organized into groups. It describes how strongly u ...
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
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.
* Population independence – If is the list x appended to itself, then ''c''
''v''() = ''c''
''v''(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''
''i'' > ''x''
''j'') without altering their rank, then ''c''
''v'' decreases and vice versa.
''c''
''v'' assumes its minimum value of zero for complete equality (all ''x''
''i'' are equal).
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).
It is, however, more mathematically tractable than the Gini coefficient.
As a measure of standardisation of archaeological artefacts
Archaeologists often use CV values to compare the degree of standardisation of ancient artefacts. Variation in CVs has been interpreted to indicate different cultural transmission contexts for the adoption of new technologies. Coefficients of variation have also been used to investigate pottery standardisation relating to changes in social organisation. Archaeologists also use several methods for comparing CV values, for example the modified signed-likelihood ratio (MSLR) test for equality of CVs.
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
The degree Celsius is the unit of temperature on the Celsius scale (originally known as the centigrade scale outside Sweden), one of two temperature scales used in the International System of Units (SI), the other being the Kelvin scale. The ...
and
Fahrenheit
The Fahrenheit scale () is a temperature scale based on one proposed in 1724 by the physicist Daniel Gabriel Fahrenheit (1686–1736). It uses the degree Fahrenheit (symbol: °F) as the unit. Several accounts of how he originally defined his ...
(both relative units, where
kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
and
Rankine scale
The Rankine scale () is an absolute scale of thermodynamic temperature named after the University of Glasgow engineer and physicist Macquorn Rankine, who proposed it in 1859.
History
Similar to the Kelvin scale, which was first proposed in 1848 ...
are their associated absolute values):
Celsius:
, 10, 20, 30, 40
Fahrenheit:
2, 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:
73.15, 283.15, 293.15, 303.15, 313.15
Rankine:
91.67, 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%.
Mathematically speaking, the coefficient of variation is not entirely linear. That is, for a random variable
, the coefficient of variation of
is equal to the coefficient of variation of
only when
. In the above example, Celsius can only be converted to Fahrenheit through a linear transformation of the form
with
, whereas Kelvins can be converted to Rankines through a transformation of the form
.
Distribution
Provided that negative and small positive values of the sample mean occur with negligible frequency, the
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
of the coefficient of variation for a sample of size
of i.i.d. normal random variables has been shown by Hendricks and Robey to be
where the symbol
indicates that the summation is over only even values of
, i.e., if
is odd, sum over even values of
and if
is even, sum only over odd values of
.
This is useful, for instance, in the construction of
hypothesis test
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
s or
confidence interval
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
s.
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).
[Lehmann, E. L. (1986). ''Testing Statistical Hypothesis.'' 2nd ed. New York: Wiley.] "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 moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant ...
s are similar ratios,
where
is the ''k''
th moment about the mean, which are also dimensionless and scale invariant. The
variance-to-mean ratio
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a prob ...
,
, is another similar ratio, but is not dimensionless, and hence not scale invariant. See
Normalization (statistics)
In statistics and applications of statistics, normalization can have a range of meanings. In the simplest cases, normalization of ratings means adjusting values measured on different scales to a notionally common scale, often prior to averaging ...
for further ratios.
In
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
, particularly
image processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
, the
reciprocal
Reciprocal may refer to:
In mathematics
* Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal''
* Reciprocal polynomial, a polynomial obtained from another pol ...
ratio
(or its square) is referred to as the
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to the noise power, often expressed in deci ...
in general and
signal-to-noise ratio (imaging)
Signal-to-noise ratio (SNR) is used in imaging to characterize image quality. The sensitivity of a (digital or film) imaging system is typically described in the terms of the signal level that yields a threshold level of SNR.
Industry stand ...
in particular.
Other related ratios include:
*
Efficiency
Efficiency is the often measurable ability to avoid wasting materials, energy, efforts, money, and time in doing something or in producing a desired result. In a more general sense, it is the ability to do things well, successfully, and without ...
,
*
Standardized moment
In probability theory and statistics, a standardized moment of a probability distribution is a moment (often a higher degree central moment) that is normalized, typically by a power of the standard deviation, rendering the moment scale invariant ...
,
*
Variance-to-mean ratio
In probability theory and statistics, the index of dispersion, dispersion index, coefficient of dispersion, relative variance, or variance-to-mean ratio (VMR), like the coefficient of variation, is a normalized measure of the dispersion of a prob ...
(or relative variance),
*
Fano factor In statistics, the Fano factor, like the coefficient of variation, is a measure of the statistical dispersion, dispersion of a probability distribution of a Fano noise. It is named after Ugo Fano, an Italian American physicist.
The Fano factor is d ...
,
(windowed VMR)
See also
*
Omega ratio The Omega ratio is a risk-return performance measure of an investment asset, portfolio, or strategy. It was devised by Con Keating and William F. Shadwick in 2002 and is defined as the probability weighted ratio of gains versus losses for some thres ...
*
Sampling (statistics)
In statistics, quality assurance, and survey methodology, sampling is the selection of a subset (a statistical sample) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt ...
*
Sharpe ratio
In finance, the Sharpe ratio (also known as the Sharpe index, the Sharpe measure, and the reward-to-variability ratio) measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its ...
*
Variance function
In statistics, the variance function is a smooth function which depicts the variance of a random quantity as a function of its mean. The variance function is a measure of heteroscedasticity and plays a large role in many settings of statisti ...
References
External links
cvequality R package to test for significant differences between multiple coefficients of variation
{{DEFAULTSORT:Coefficient Of Variation
Statistical deviation and dispersion
Statistical ratios
Income inequality metrics