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 $$ x_i $$, $$i = 1, 2,\ldots, n$$ 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} $$

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 .