In probability theory and statistics, the coefficient of variation is a normalized measure
of dispersion of a probability distribution or frequency distribution.
It is defined as the ratio of the standard deviation to the mean . It is also known as
unitized risk or the variation coefficient.
The absolute value of the CV is sometimes known as relative standard deviation, which
is expressed as a percentage.
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 mean of the population.
The coefficient of variation should be computed only for data measured on a ratio scale, as
these are 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 are interval scales that can take both positive and negative
values, whereas the Kelvin scale has an absolute null value, and negative values are nonsensical.
Hence, the Kelvin scale is a ratio scale.
While the standard deviation can be derived on both the Kelvin and the Celsius scale,
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 would
vary depending on the expected value of measurements.
This is the case for laboratory values that are measured based on chromatographic methods.
A nonparametric possibility is the quartile coefficient of dispersion, i.e. interquartile
range divided by the median . 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 In many applications, it can be assumed that
data are log-normally distributed.
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(RMSD).
Essentially the CV(RMSD) 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.
Laboratory measures of intra 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.
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
where the symbol 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, Lehmann "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, , which
are also dimensionless and scale invariant.
The variance-to-mean ratio, , is another similar ratio, but is not dimensionless, and hence
not scale invariant.
See Normalization for further ratios.
In signal processing, particularly image processing, the reciprocal ratio is referred to as the
signal to noise ratio.
Relative standard deviation, Efficiency,
Standardized moment, Variance-to-mean ratio,
Fano factor, Signal-to-noise ratio,
Signal-to-noise ratio
See also Sampling