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

[math]\displaystyle{ K = \left( 1 + \frac{1}{c_v^2} \right) \ \frac{(n - 1) \ \hat{c}_v^2}{1 + (n - 1) \ \hat{c}_v^2/n} }[/math]

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When [math]\displaystyle{ c_v }[/math] is smaller than 1/3, then [math]\displaystyle{ K }[/math] is approximately chi-square distributed with [math]\displaystyle{ n - 1 }[/math] degrees of freedom. In the original article by McKay, the expression for [math]\displaystyle{ K }[/math] looks slightly different, since McKay defined [math]\displaystyle{ \sigma^2 }[/math] with denominator [math]\displaystyle{ n }[/math] instead of [math]\displaystyle{ n - 1 }[/math]. McKay's approximation, [math]\displaystyle{ K }[/math], for the coefficient of variation is approximately chi-square distributed, but exactly noncentral beta distributed .