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.^[1] Statistical methods for the coefficient of variation often utilizes McKay's approximation.^[2][3][4][5]

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

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

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When ${\displaystyle c_{v}}$ is smaller than 1/3, then ${\displaystyle K}$ is approximately chi-square distributed with ${\displaystyle n-1}$ degrees of freedom. In the original article by McKay, the expression for ${\displaystyle K}$ looks slightly different, since McKay defined ${\displaystyle \sigma ^{2}}$ with denominator ${\displaystyle n}$ instead of ${\displaystyle n-1}$. McKay's approximation, ${\displaystyle K}$, for the coefficient of variation is approximately chi-square distributed, but exactly noncentral beta distributed .^[6]

References

1. McKay, A. T. (1932). "Distribution of the coefficient of variation and the extended "t" distribution". Journal of the Royal Statistical Society. 95: 695–698. doi:10.2307/2342041.
2. Iglevicz, Boris; Myers, Raymond (1970). "Comparisons of approximations to the percentage points of the sample coefficient of variation". Technometrics. 12 (1): 166–169. doi:10.2307/1267363. JSTOR 1267363.
3. Bennett, B. M. (1976). "On an approximate test for homogeneity of coefficients of variation". Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl. 22: 169–171.
4. Vangel, Mark G. (1996). "Confidence intervals for a normal coefficient of variation". The American Statistician. 50 (1): 21–26. doi:10.1080/00031305.1996.10473537. JSTOR 2685039..
5. Forkman, Johannes. "Estimator and tests for common coefficients of variation in normal distributions" (PDF). Communications in Statistics - Theory and Methods. pp. 21–26. doi:10.1080/03610920802187448. Retrieved 2013-09-23.
6. Forkman, Johannes; Verrill, Steve. "The distribution of McKay's approximation for the coefficient of variation" (PDF). Statistics & Probability Letters. pp. 10–14. doi:10.1016/j.spl.2007.04.018. Retrieved 2013-09-23.