McKay's approximation for the coefficient of variation

In statistics, McKay's approximation of the coefficient of variation is an estimate based on a sample from a normally distributed population. It was introduced in 1932 by A. T. McKay Under certain reasonable conditions, it has a chi-square distribution.

Let $x_{i}$, $i=1,2,\ldots ,n$ be $n$ independent observations from a $N(\mu ,\sigma ^{2})$ normal distribution. The population coefficient of variation is $c_{v}=\sigma /\mu$. Let ${\bar {x}}$ and $s\,$ denote the sample mean and the sample standard deviation, respectively. Then ${\hat {c}}_{v}=s/{\bar {x}}$ is the sample coefficient of variation. McKay’s approximation is

$K=\left(1+{\frac {1}{c_{v}^{2}}}\right)\ {\frac {(n-1)\ {\hat {c}}_{v}^{2}}{1+(n-1)\ {\hat {c}}_{v}^{2}/n}}$

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