In statistics, McKay's approximation of the coefficient of variation is a statistic based on a sample from a normally distributed population. It was introduced in 1932 by A. T. McKay. Statistical methods for the coefficient of variation often utilizes McKay's approximation.
Let
x
i
,
i
=
1
,
2
,
…
,
n
be
n
independent observations from a
N
(
μ
,
σ
2
)
normal distribution. The population coefficient of variation is
c
v
=
σ
/
μ
. Let
x
¯
and
s
denote the sample mean and the sample standard deviation, respectively. Then
c
^
v
=
s
/
x
¯
is the sample coefficient of variation. McKay’s approximation is
K
=
(
1
+
1
c
v
2
)
(
n
−
1
)
c
^
v
2
1
+
(
n
−
1
)
c
^
v
2
/
n
Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When
c
v
is smaller than 1/3, then
K
is approximately chi-square distributed with
n
−
1
degrees of freedom. In the original article by McKay, the expression for
K
looks slightly different, since McKay defined
σ
2
with denominator
n
instead of
n
−
1
. McKay's approximation,
K
, for the coefficient of variation is approximately chi-square distributed, but exactly noncentral beta distributed .
