EASY STATISTICAL TESTS 103 explanation involving chance is quite unlikely, and he then has the option of claiming that a real difference exists. The fact that he decides to make such a claim obviously does not rule out the possibility that there really is no difference. In each of the first two tests, the probabili- ties given in the tables are based upon the assumption that a single com- parison only is to be made. Suppose, however, that 100 such pairs of observed differences are being tested and that, unknown to the investi- gator, none of them is actually real. In such a ease, there will neverthe- less be a number of instances to which the test will assign a low prob- ability. If the experimenter decides to claim that those differences assigned a low probability are real, he will be wrong. Another way of stating this is to say that when more than one pair of averages is being compared by means of a statistical test, the true probability of each difference will be greater than that found, and the experimenter's risk in making a decision to take some action will be correspondingly increased. The Multiple Rank Test is designed to handle problems involving more than one difference between means, and it does so by requiring that the differences be larger than would otherwise be the case before a given probability can be claimed. The size of the difference required for a certain probability of the null hypothesis will increase as the number of treatments among which all pairs are to be compared increases. Suppose that, as in the previous example, antiperspirants are being tested, but that there are four materials involved, and that it has already been established that each is effective by comparing it with a control as described above. It is now of interest to learn whether any of the four materials is more effective than any other. This time, a decision is made to utilize the skin of the subject's backs, so that all four materials may be applied to each of the subjects in the test, thus giving a "paired" pro- cedure analogous to the two-sample paired test. Here, although the experimenter may feel that there is a difference between the action of an antiperspirant in an axilla compared to the use of the same material on the skin of a subject's back, it is assumed that he believes the relative effectiveness of the four materials will not be changed by the use of backs rather than axillae. In carrying out this experiment, it is advisable to observe a well- established experimental design principle to avoid any likelihood of a systematic effect related to location on the back (3). This is done by assigning each treatment to a randomly-selected spot on each back, with a different random selection for each subject. As before, some measure of the quantity of moisture generated on each site, under standard condi-

104 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Table V Results of Multiple Antiperspirant Test Weight Gain, Mg Subject Formula Formula Formula Formula No. A B C D 1 250 304 177 408 2 147 329 98 573 3 362 487 85 319 4 103 675 299 274 5 201 278 311 585 6 177 190 112 612 7 96 453 136 899 Sums 1336 2716 1218 3670 Averages 191 388 174 524 Table VI Ranking of Antipcrspirant Test Data Multiple Rank Test Sub- ject For•nula Formula Formula Formula No. A Rank B Rank C Rank D Rank 1 250 2 304 3 177 1 408 4 2 147 2 329 3 98 1 573 4 3 362 2 487 3 85 1 319 4 4 103 i 675 4 299 2 274 3 5 201 i 278 2 311 3 585 4 6 177 2 190 3 112 I 612 4 7 96 i 453 3 136 2 899 4 Rank Sums 11 21 11 27 Rank differences (ignoring signs): A -- B = 10 (A is less than B) A -- C = 0 (A and C are the same ) A -- D = 16 (A is less than D) B -- C = 10 (C is less than B) B -- D = 6 (B is less than D) C -- D = 16 (C is less than D) tions, is obtained. Table V shows the results of the test, using seven subjects. .Each of the four sets of data is now ranked within each subject-- that is, for each subject the four different formulations are ranked. In case of ties, the same procedure as described previously is used. After ranking the data, the column of seven rank numbers for each of the four materials is added (Table VI). Differences between every possible