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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Coefficient of variation | 1/3 | https://en.wikipedia.org/wiki/Coefficient_of_variation | reference | science, encyclopedia | 2026-05-05T07:23:32.496013+00:00 | kb-cron |
In probability theory and statistics, the coefficient of variation (CV), also known as normalized root-mean-square deviation (NRMSD), and relative standard deviation (RSD), is a standardized measure of dispersion of a probability distribution or frequency distribution. It is defined as the ratio of the standard deviation
σ
{\displaystyle \sigma }
to the mean
μ
{\displaystyle \mu }
(or its absolute value,
|
μ
|
{\displaystyle |\mu |}
), and often expressed as a percentage ("%RSD"). 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 (CV) is defined as the ratio of the standard deviation
σ
{\displaystyle \sigma }
to the mean
μ
{\displaystyle \mu }
,
C
V
=
σ
μ
.
{\displaystyle CV={\frac {\sigma }{\mu }}.}
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) 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. 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 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 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
(
Q
3
−
Q
1
)
/
2
{\displaystyle {(Q_{3}-Q_{1})/2}}
divided by the average of the quartiles (the midhinge),
(
Q
1
+
Q
3
)
/
2
{\displaystyle {(Q_{1}+Q_{3})/2}}
. 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.
== Examples == In the examples below, we will take the values given as randomly chosen from a larger population of values.
The data set [100, 100, 100] has constant values. Its standard deviation is 0 and average is 100, giving the coefficient of variation as 0 / 100 = 0 The data set [90, 100, 110] 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 [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 = 1.18 In these examples, we will take the values given as the entire population of values.
The data set [100, 100, 100] has a population standard deviation of 0 and a coefficient of variation of 0 / 100 = 0 The data set [90, 100, 110] has a population standard deviation of 8.16 and a coefficient of variation of 8.16 / 100 = 0.0816 The data set [1, 5, 6, 8, 10, 40, 65, 88] 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
s
{\displaystyle s\,}
to the sample mean
x
¯
{\displaystyle {\bar {x}}}
:
c
v
^
=
s
x
¯
{\displaystyle {\widehat {c_{\rm {v}}}}={\frac {s}{\bar {x}}}}
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:
c
v
^
∗
=
(
1
+
1
4
n
)
c
v
^
{\displaystyle {\widehat {c_{\rm {v}}}}^{*}={\bigg (}1+{\frac {1}{4n}}{\bigg )}{\widehat {c_{\rm {v}}}}}
=== 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:
c
v
^
r
a
w
=
e
s
ln
2
−
1
{\displaystyle {\widehat {cv}}_{\rm {raw}}={\sqrt {\mathrm {e} ^{s_{\ln }^{2}}-1}}}
where
s
ln
{\displaystyle {s_{\ln }}\,}
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
s
b
{\displaystyle s_{b}\,}
is converted to base e using
s
ln
=
s
b
ln
(
b
)
{\displaystyle s_{\ln }=s_{b}\ln(b)\,}
, and the formula for
c
v
^
r
a
w
{\displaystyle {\widehat {cv}}_{\rm {raw}}\,}
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:
G
C
V
K
=
e
s
ln
−
1
{\displaystyle \mathrm {GCV_{K}} ={\mathrm {e} ^{s_{\ln }}\!\!-1}}