McKay's approximation for the coefficient of variation

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 $x_i$, $i=1,2,\dots ,n$ be $n$ independent observations from a $N(\mu ,{\sigma }^2)$ normal distribution. The population coefficient of variation is $c_v=\sigma /\mu$. Let $\overline{x}$ and $s$ denote the sample mean and the sample standard deviation, respectively. Then ${\hat{c}}_v=s/\overline{x}$ is the sample coefficient of variation. McKay's approximation is

$K=(1+\frac{1}{c_v^2})\frac{(n-1){\hat{c}}_v^2}{1+(n-1){\hat{c}}_v^2/n}$

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 ${\sigma }^2$ 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 .