14 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Coefficient of variation | 3/3 | https://en.wikipedia.org/wiki/Coefficient_of_variation | reference | science, encyclopedia | 2026-05-05T07:23:32.496013+00:00 | kb-cron |
== 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%. Mathematically speaking, the coefficient of variation is not entirely linear. That is, for a random variable
X
{\displaystyle X}
, the coefficient of variation of
a
X
+
b
{\displaystyle aX+b}
is equal to the coefficient of variation of
X
{\displaystyle X}
only when
b
=
0
{\displaystyle b=0}
. In the above example, Celsius can only be converted to Fahrenheit through a linear transformation of the form
a
x
+
b
{\displaystyle ax+b}
with
b
≠
0
{\displaystyle b\neq 0}
, whereas Kelvins can be converted to Rankines through a transformation of the form
a
x
{\displaystyle ax}
.
== 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
{\displaystyle n}
of i.i.d. normal random variables has been shown by Hendricks and Robey to be
d
F
c
v
=
2
π
1
/
2
Γ
(
n
−
1
2
)
exp
(
−
n
2
(
σ
μ
)
2
⋅
c
v
2
1
+
c
v
2
)
c
v
n
−
2
(
1
+
c
v
2
)
n
/
2
∑
∑
′
i
=
0
n
−
1
(
n
−
1
)
!
Γ
(
n
−
i
2
)
(
n
−
1
−
i
)
!
i
!
⋅
n
i
/
2
2
i
/
2
⋅
(
σ
μ
)
i
⋅
1
(
1
+
c
v
2
)
i
/
2
d
c
v
,
{\displaystyle \mathrm {d} F_{c_{\rm {v}}}={\frac {2}{\pi ^{1/2}\Gamma {\left({\frac {n-1}{2}}\right)}}}\exp \left(-{\frac {n}{2\left({\frac {\sigma }{\mu }}\right)^{2}}}\cdot {\frac {{c_{\rm {v}}}^{2}}{1+{c_{\rm {v}}}^{2}}}\right){\frac {{c_{\rm {v}}}^{n-2}}{(1+{c_{\rm {v}}}^{2})^{n/2}}}\sideset {}{^{\prime }}\sum _{i=0}^{n-1}{\frac {(n-1)!\,\Gamma \left({\frac {n-i}{2}}\right)}{(n-1-i)!\,i!\,}}\cdot {\frac {n^{i/2}}{2^{i/2}\cdot \left({\frac {\sigma }{\mu }}\right)^{i}}}\cdot {\frac {1}{(1+{c_{\rm {v}}}^{2})^{i/2}}}\,\mathrm {d} c_{\rm {v}},}
where the symbol
∑
∑
′
{\textstyle \sideset {}{^{\prime }}\sum }
indicates that the summation is over only even values of
n
−
1
−
i
{\displaystyle n-1-i}
, i.e., if
n
{\displaystyle n}
is odd, sum over even values of
i
{\displaystyle i}
and if
n
{\displaystyle n}
is even, sum only over odd values of
i
{\displaystyle 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 === Liu (2012) reviews methods for the construction of a confidence interval for the coefficient of variation. Notably, Lehmann (1986) derived the sampling distribution for the coefficient of variation using a non-central t-distribution to give an exact method for the construction of the CI.
== Similar ratios == Standardized moments are similar ratios,
μ
k
/
σ
k
{\displaystyle {\mu _{k}}/{\sigma ^{k}}}
where
μ
k
{\displaystyle \mu _{k}}
is the kth moment about the mean, which are also dimensionless and scale invariant. The variance-to-mean ratio,
σ
2
/
μ
{\displaystyle \sigma ^{2}/\mu }
, 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
μ
/
σ
{\displaystyle \mu /\sigma }
(or its square) is referred to as the signal-to-noise ratio in general and signal-to-noise ratio (imaging) in particular. Other related ratios include:
Efficiency,
σ
2
/
μ
2
{\displaystyle \sigma ^{2}/\mu ^{2}}
Standardized moment,
μ
k
/
σ
k
{\displaystyle \mu _{k}/\sigma ^{k}}
Variance-to-mean ratio (or relative variance),
σ
2
/
μ
{\displaystyle \sigma ^{2}/\mu }
Fano factor,
σ
W
2
/
μ
W
{\displaystyle \sigma _{W}^{2}/\mu _{W}}
(windowed VMR) Second-order coefficient of variation
V
2
=
σ
2
σ
2
+
μ
2
{\displaystyle V_{2}={\sqrt {\frac {\sigma ^{2}}{\sigma ^{2}+\mu ^{2}}}}}
which is bounded between 0 (no variance) and 1 (maximal relative variance, i.e.
σ
≫
μ
{\displaystyle \sigma \gg \mu }
).
== See also == Standard score Information ratio Omega ratio Sampling (statistics) Variance function
== References ==
== External links == cvequality: R package to test for significant differences between multiple coefficients of variation