JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 3.0 2.0 or be tabulated as follows ß Fig. I. Number of positive reactions Number of positive reactiota Cumulative number at or below wlMch the product at or above which the product of subjects tested can be accepted can be rejected .. . [No decision to accept can be] 1-22 •reached before 485 tests have• 1 23-484 (been carried out. J 2 485-533 0 2 534-994 0 3 995-1,044 1 3 1,045- 1 4 For less than 485 subjects tested the product cannot be accepted, since this requires a negative amount of reiects however, at 485 tests 0 rejects indicates that the product is acceptable. For any number of tests from 23 to 533, 1 reject implies continuance of testing, and 2 rejects implies rejection of the product. Since the number of subjects to be tested is not fixed, it is useful to calcu- late the average number of tests which can be expected to be necessary. This number depends on the actual proportion of positives which the product will give rise to in the whole population from which the sample has been taken, and it is conveniently shown in a graph drawn from a few fixed points calculated in accordance with the following tabulated formulae: 230

STATISTICAL METHODS IN THE COSMETIC INDUSTRY Fraction giving positives 0 p• s P2 1 Averagesample h•ls (h•--o•(h•+h2))/ h•h2ls(1--s) (h•--[3(h•+h•))! h•[(1--s) number (s--p•) (p•--s) _ These points form the basis of the curve to be drawn showing the relation between average number to be inspected and the true per cent positive reactions in the population. We have ' Fraction giving positives ...... 0 0.0005 0.00196 0.005 1.000 Per cent giving positives ...... 0 0.05 0-196 0.5 100.0 Average number of tests to be carried out before reaching a conclusion ... 485 522 464 251 0-95 giving the graph' 5OO •00 o • g •oo •o o.[ o.• o.N 04 o.• o.'• 'o.• o.• o.• ACTUAL • pOSITIVE REACTIONS Y•HICH "OULD BE OBTAINED IF THE •'oedOLE POPULATION •ERE EXAMINED. Fig. 2. 231