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 , be independent observations from a normal distribution. The population coefficient of variation is . Let and denote the sample mean and the sample standard deviation, respectively. Then is the sample coefficient of variation. McKay’s approximation is

Note that in this expression, the first factor includes the population coefficient of variation, which is usually unknown. When is smaller than 1/3, then is approximately chi-square distributed with degrees of freedom. In the original article by McKay, the expression for looks slightly different, since McKay defined with denominator instead of . McKay's approximation, , 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.
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