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.{{cite journal | last1 = McKay | first1 = A. T. | year = 1932 | title = Distribution of the coefficient of variation and the extended "t" distribution | journal = Journal of the Royal Statistical Society | volume = 95 | pages = 695–698 | doi=10.2307/2342041}} Statistical methods for the coefficient of variation often utilizes McKay's approximation.{{cite journal |jstor=1267363 |title=Comparisons of approximations to the percentage points of the sample coefficient of variation |year=1970 |last1=Iglevicz |first1=Boris |last2=Myers |first2=Raymond |journal=Technometrics |volume=12 |issue=1 |pages=166–169|doi = 10.2307/1267363 }}{{cite journal | last1 = Bennett | first1 = B. M. | year = 1976 | title = On an approximate test for homogeneity of coefficients of variation | journal = Contributions to Applied Statistics Dedicated to A. Linder. Experentia Suppl | volume = 22 | pages = 169–171 }}{{cite journal |jstor=2685039 |title=Confidence intervals for a normal coefficient of variation |year=1996 |last1=Vangel |first1=Mark G. |journal=The American Statistician |volume=50 |issue=1 |pages=21–26 |doi=10.1080/00031305.1996.10473537}}.{{cite web| url=http://pub.epsilon.slu.se/4489/1/forkman_j_110214.pdf | title=Estimator and tests for common coefficients of variation in normal distributions | access-date=2013-09-23 |last1=Forkman |first1=Johannes |journal=Communications in Statistics - Theory and Methods |volume=38 |pages=21–26 | doi=10.1080/03610920802187448}}

Let $x\_i$, $i\; =\; 1,\; 2,\backslash ldots,\; n$ be $n$ independent observations from a $N(\backslash mu,\; \backslash sigma^2)$ normal distribution. The population coefficient of variation is $c\_v\; =\; \backslash sigma\; /\; \backslash mu$. Let $\backslash bar\{x\}$ and $s\; \backslash ,$ denote the sample mean and the sample standard deviation, respectively. Then $\backslash hat\{c\}\_v\; =\; s/\backslash bar\{x\}$ 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 $\backslash sigma^2$ 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

.{{cite web| url=http://pub.epsilon.slu.se/3317/1/Forkman_Verrill_080610.pdf | title=The distribution of McKay's approximation for the coefficient of variation | access-date=2013-09-23 |last1=Forkman |first1=Johannes |last2=Verrill |first2=Steve |journal=Statistics & Probability Letters |volume=78 |pages=10–14 | doi=10.1016/j.spl.2007.04.018}}