Some properties of the sample coefficient of variation

In this dissertation some properties of the sample coefficient of variation are investigated and the results are summarized below.

The exact frequency function of the S.C.V., whose parent distribution is assumed to be normal, is derived. The probability of negative values of the S.C.V. is discussed and the moments of the S.C.V. are shown not to exist. The above mentioned frequency function is then factored into a form suitable for computer calculations and an exact table of percentage points is derived and given in the Appendix.

All moments of the S.C.V. whose parent population is restricted to the non-negative range are shown to exist. Upper bounds for such moments are given and these upper bounds are used to show that the moments of the S.C.V. uniquely determine their distribution. Cramer also shows that this distribution is asymptotically normal.

Using Fisher's k statistics approximations to the mean and next three central moments of the S.C.V., are obtained through the 1/n³ terms. Investigations for the normal and gamma parent cases are then undertaken. The approximate moments of the Statistic Z = S – t₀ X̄ are also derived and several of its properties are then studied.

Several approximations to the distribution of the S.C.V. are investigated. Tables of their percentage points are then derived and compared with the previously obtained exact results. It is found that for small values of the population coefficient of variation (C.V.), McKay's approximation gives excellent results. This approximation becomes worse as the C.V. increases. For intermediate sample sizes a Pearson fit to the approximate moments of the S.C.V. gives very good results. For large sample sizes. say greater than 100 the asymptotic normality approximation gives adequately accurate percentage points. Surprisingly, the latter approximation is more accurate than the one which assumes normality of the S.C.V. and mean and variance given to (1/n³). Also included are several tables which summarize the degrees of accuracy of the above approximations.

A more thorough investigation into the asymptotic normality. of the S.C.V. is undertaken. Also, there is investigated the robustness with respect to changes of the parent populations, of the percentage points of the S.C.V. For this purpose the effects, on the percentage points, of replacing the normal parent assumption by a gamma distribution is investigated. For small values of the C.V. the robustness is found to be exceptionally good. As the C.V. increased, larger sample sizes are found to be necessary to achieve a robustness of high quality. This situation is again illustrated by a table.

For values of the C.V. less than .5 the above properties of the S.C.V., also hold for the non-central t distribution. In fact, the exact tables derived in this dissertation can be used to find the exact percentage points for small values of the C.V. of the non-central t.

The S.C.V. is then generalized to a form suitable for handling certain problems in the analysis of variance. The exact percentage points of this generalized sample coefficient of variation (G.S.C.V.) can be found by the use of the previously derived percentage points for the S.C.V.

The above results are illustrated by several numerical examples. These examples describe the application of the S.C.V. to testing statistical hypothesis, finding the power curves of such tests; obtaining confidence intervals for the C.V. It is also used to study certain quality control problems and in the application of the G.S.C.V. to the analysis of experimental data.