JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS Lot tolerance per cent defective 0.5 per cent. Process average per cent defective 0.05 per cent. i 1st sample inspected 2nd sample inspected -- No. of positives Number of posi- Maximum aver- Size of which if exceed- Size of rives from both age % reactions Lot size sample ed require exam- sample 1st q- 2nd sam- given by whole ination of a sec- pies not to be population ond sample exceeded -- 301-400 275 0 -- --- 0.04 401-500 340 0 110 1 O. C'4 501-600 355 0 140 1 0.05 601-800 400 0 185 I 0.06 801-1,000 430 0 200 1 0.07 1,001-2,000 490 0 265 1 0.08 2,001-3,000 520 0 290 1 0.09 3,001-4,000 530 0 570 2 0.11 4,001-5,000 540 0 580 2 0.11 5,001•-7,000 545 0 615 2 0.11 7,001-10,000 550 0 620 2 0'12 10,001-20,000 555 0 925 3 0.13 20,001-50,000 560 0 940 3 O. 14 , 50,001-100,000 560 0 1,210 4 O. 15 By adopting this scheme we have reduced the sampling for our 10,000 lot from 760 people in one test--rejecting the product if 2 or more positive reactions are found--to 550 people, and if one or two positives are obtained we inspect a further 620, rejecting if three or more positives are found in the 550 plus 620--equals 1,170--sample. Only rarely will it be considered financially worth while to carry out so many tests. SEQUENTIAL ANALYSIS During the War the Statistical Research Group of Columbia University 8 developed this sample inspection method in order to reduce the size of the samples inspected to as small a number as possible. Samples are inspected for defects and the cumulative number of defects in the samples either plotted in a graph or noted in a table between two columns. When the cumulative number of rejects is reached which exceeds the limit number for the size of sample, the lot is rejected. When, for the size of sample, it is less than another limit number, the lot is accepted. Thus a final size of sample is not used, and conclusions are enabled to be drawn as the test proceeds, resulting in the smallest possible amount of inspection. Such a system might be of use in deciding whether a product was likely to give rise to many cases of dermatitis. It is necessary first to decide on four quantities: these are the fraction of positive reactions below which the ' product will be considered suitable for marketing (= p•) the fraction of 228

.•jZ•?:::: STATISTICAL METHODS IN THE COSMETIC INDUSTRY '•i'?Positive reactions arising which will be considered sufficient to reject the product as unsuitable for marketing (= p•) the probability of rejecting a i::ii?'product which is really of acceptable quality (---- a) and the probability •'•J•ii?'of accepting a product wkich is really of reject quality (= •). Using the same values as were used in our inspection of the Dodge and Romig schemes, we have' p• = 0.0005 (i.e., 0-05 per cent) p, ---- 0.005 (i.e., 0.5 per cent) a = 0-10 • =O.lO We now have first to calculate the following: G• = Iog,op,/p• G• = log•0(1 --p•)/(1 --p=) • = ZOg•o(1 - •)/• a = log•0(1 - •)/•. Note: whena=fi: a----b. ??And use these to calculate ß /•::/ • = •/(• + •) •i•:i ß • = a/(• + •) 7!::,: s = G•,/(G• q- ß , Our acceptance and rejection limits are given by the equations ß 7' For acceptance d• = sn -- h• :"•' For rejection d R = s•'• + h• Note' whena=fi h•=h• where n is the number inspected, da the decision to accept and d R the decision to reject. In our example we have ß G• -- log 0-005/0.0005 = 1.00000 G• = log 0.9995/0.995 = 0.00196 a = b = log 0.9/0-1 = 0.95424 from which h• ---- h• ---- 0.95424/1.00196 = 0.95238 s = 0.00196/1.00196 = 0'00196 This gives our acceptance line ß d A = sn -- h• = 0.00196n -- 0'95238 and rejection line ' d R = sn q- h• = 0.00196n q- 0.95238 These can either be drawn into a graph by substituting, say, n ' 0 and n = 500 to give two fixed points of the graph ß 229