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 .
