408 D. C. Cullum It can be seen that the relation works well up to sixty subjects, but for larger numbers tends to overestimate. This may be due to the very small number of tests on more than sixty subjects. The accuracy of prediction in individual tests is subject to considerable variation, however, depending as it does on how lucky one is with respect to the distri- bution of sweat weights in individual groups of subjects. BACKGROUND DATA AND THEIR IMPLICATIONS The greater part of the necessary background information came from an experiment, described earlier, in which sixteen women, who had used only an alcoholic deodorant for the preceding 2 weeks, took part in hot-room sittings on 5 consecutive days, during which they received one daily treatment with the same alcoholic deodorant on both axillae. After an interval of 2 weeks they returned and repeated the same experiment. Since all the axillae had received the same treatment it was legitimate to call any arbitrarily selected group the test group and the remainder the control group. DISTRIBUTION OF SWEAT WEIGHTS AND THEIR LOGARITHMS Since each subject yielded four pad weights, the maximum possible number of single observations was 16 x 10 x 4=640. In the event hot-room appointments were missed on three occasions, so that the actual number of pad weights recorded was 628. Figs 1 and 2 show histograms of the actual weights and their logarithms to the base 10. Clearly, the histogram of weights is positive skewed, but equally clearly, the histogram of log weights is negatively skewed. Thus, whilst it is not rigorously correct to apply to the population of weights statistical tests which depend on a Gaussian distribution, it is not rigorously correct to apply them to the population of log weights either. In practice neither population deviates very seriously from normality, and the inevitable errors will be of no great consequence whichever population is used, provided always that the distributions obtained in a particular experiment are of reasonably smooth and regular shapes. In practice, in a test on thirty or forty subjects, yielding twice that number of pad weights for each treatment, the distributions are quite often very irregular and in such IOO 90 80 '• 7o •o[- 0 -- = • ,• . . • . 7 ..... ' 6 6 6 6 6 6 6 6 6 -' Sweat weight (g) Fisure 1. DistributioD of' sweat wei8hts. SixteeD subjects x ted sittiDss x fou• pads per sittin8 tl•'CC subject-sittiDss missed total obse•,atiom, 68:2.

Rapid hot-room testing of antiperspirants 409 130 - 120 I10 •00 90 •, 80 '• 60 z 50 4o 30 zo I0 0 oo g8 88 8• • Log•o sweol weigh• •ases the sample standard deviation will necessarily be an unreliable estimate of the population standard deviation, regardless of whether the actual weights or their logarithms are used. Ove• a large number of tests we have found that the petGentage teduGtions in sweat weight calculated f•om geometric and arithmetic means tardy dJffe• by more than 2 or •, and deGisions based on the t•sults would be the same whJGhevet value was used. M•jor differences have been few, and have always proved to indicate a peculia• distri- bution of either test or control sweat weights. It •oDld •m pNdent to GoDsider the ptaGtiGal advantages as well as th• mathe- matical rigour of using arithmetic or geometric means. Wooding (?) has shown that the distribution of log weights deviates less from normality than the distribution of the weights themselves, and has argued that it is therefore more correct to use geometric means. With present levels of ptoduGt petfotmanGe there or disadvantage in doing so. If tD• tim• •v• orals, ho•v•, •D•D p•odD t• •ill •dD • the sweat weight in at least a few problem. Since the geometric mean of any array of numbers which includes zero is zero, •dhetenGe to the USe Of geometdG meads would then either lead to the manifestly absurd GODG]DSJOD t•t ]00• t•dDGtJOD •d b•D •G•v•d W•D OD][ OD• t•St •i]]• yielded zero weight, or require rejection of the ve• results which most strikingly demonstrated the success of the p•oduct. This m•y be in the far distant future, but neverthdess is food for thought. RATIOS The same data were used to study the variation between subjects and within subjects of the ratio of sweat weights from right and left sides. The object was to discover whether this ratio was sufficiently constant in the absence of antiperspirant treatment to serve as a parameter for the measurement of efficacy. Presentation of full individual results would involve tables containing several hundred figures and it is hoped that the following digest will sufficiently illustrate the main points.