98 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table Ii Ranking o[ Viscosity Test Data Old Formula New Formula Value Rank Value Rank 1558 7 1560 8 1600 10.5 1423 5 1423 5 1300 2 1842 15 1600 10.5 1761 14 1329 3 1602 12 1590 9 1657 13 1423 5 1298 I Rank totals 76.5 43.5 T = 76.5 = sum of ranks in smaller sample. Table II shows the data of Table I after completion of these steps. For clarity, both columns of ranks have been added. The number of items in the smaller group is seven, and this is called M. That in the larger is M q- 1 or eight, and this is referred to as N. The rank number, T, for the 3/items is 76.5. The three numbers M, N, and T are used to locate critical values in Table A (see Appendix). Table A-1 gives lower critical values of T of 41, 39, 36 and 34, for the four probabilities 0.05, 0.025, 0.01 and 0.005, while Table A-2 gives upper critical values of 71, 73, 76 and 78 for the same probabilities. In order to claim "significance" for the difference between the two averages, the calculated value of T must either exceed one of the upper critical values or be less than one of the lower values (in the case of equal sample sizes, these criteria may be applied to either of the two rank totals). In the present case, the calculated value is 76.5, which lies between 76 and 78 in Table A-2, and is therefore associated with a probability between 0.01 and 0.005. The investigator therefore concludes that the likelihood of obtaining a total this large by chance (if the two formulas really have the same viscosity and the observed difference is due simply to sampling and experimental variation) lies between 0.5 and 1%. This represents a fairly low probability, and he would probably be more willing to accept the interpretation that a real difference exists than that such an unlikely chance event occurred. There is a technicality applying to this conclusion, connected with the use of the words "one-tailed" in the titles for Tables A-1 and A-2. In the example, it was assumed that the question being asked of the

EASY STATISTICAL TESTS 99 data was, "Does the new formula give product with a lower viscosity than the old one ?" If the question had been, "Are the viscosities differ- ent?" (regardless of which was lower), a "two-tailed" test would have been required, and the tabular value of T would have needed to be higher for the upper limit and lower for the lower limit for any given probability. Both the one- and two-tailed tables appear in expanded form in the literature (1). Another important point connected with interpretation of this example should be mentioned, and applies to most statistical significance tests. If the user fails to obtain a low and convincing probability, he should not claim that the two means being compared are not different, but should conclude only that he has been unable to demonstrate a differ- ence to his satisfaction. The Wilcoxon Rank Sum Test (1) may be used for almost any nu- merical data, whether they are viscosities, weights, percentages or ratios, arbitrary scores, or the like. The only major requirement to assure reasonable validity is that the investigator be able to assume without much doubt that the experimental or measurement errors applying to the two sets of data are of the same magnitude. This would be the normal situation if all conditions surrounding the two sets of tests were similar (same technician, similar sampling procedures, etc.). WILCOXON SIGNED-RANK TEST FOR PAIRED DATA In contrast to the foregoing example, in which each of the 15 ob- servationswere independent of each other, it maybe possible to "pair" the data so that the tests in each of the two treatments are somehow associated. If this can be done, the ability of the experiment to show a real difference is often enhanced. When such paired data are avail- able, the test described for the first section is still valid, but a more sensi- tive procedure is available. As an example of paired data, suppose that an antiperspirant is being tested by applying it to one axilla of each of several subjects, leaving the other axilla untreated for comparison. The question to be an- swered is whether antiperspirant action may be claimed for the product, and, if so, the magnitude of the probability of error in making such a claim. The perspiration generated in each axilla under standard condi- tions is measured, and a figure representing the total milligrams of moisture found in each case is obtained (treatment vs. control for each subject). To avoid any possible bias, any left-right differences which may exist are "balanced out" in accordance with good experimental