28 BELL SYSTEM TECHNICAL JOURNAL 



This article will be devoted to the type of problem termed "Sam- 

 pling of Attributes." ^ In it are included results from an extensive 

 series of computations in the form of charts which may be of value 

 in the solution of practical engineering problems. The nomenclature 

 is general, so as to be applicable to a wide variety of practical problems. 

 For convenience in discussion we shall divide the units of any sample 

 into the two mutually exclusive classes, "defective" and "satis- 

 factory." The following notation will be used: 

 N = total number of items in universe, 

 n = total number of items in sample, 

 X = number of defective items in universe (unknown), 

 c == number of defective items in sample (observed), 

 u'(X) = a priori probability that the universe will contain 

 exactly X defectives, 

 W{Xi, Xo) — a posteriori probability that the universe contains a 

 number of defectives X such that Xi = X ^ X2. 



It is of extreme importance that, at the outset, the significance of 

 the symbol w{X) in sampling problems be clearly defined. It is a 

 measure of the probability, before the sample is taken, that the lot or 

 universe in question contains X defective items and N — X satis- 

 factory items. It may be based on previous samples, or the reputation 

 of the manufacturer producing those items, or on any one or more 

 of a number of other pertinent data. For example, even before a 

 sampling inspection, we should unhesitatingly say that in a lot of 

 1,000 relays sent out by a reputable manufacturer it is very much 

 more likely, a priori, that the lot will contain less than 100 relays 

 with a short-circuited winding than that the lot will contain more 

 than 800 relays defective in the same respect. We should probably 

 find ourselves in a quandary, however, if we attempted to state 

 without a sample inspection, the relative likelihoods of 3, 4, 5, 6, 

 •••, etc., defectives existing in the lot. w{X) is a function whose 

 numerical value is assumed to state this a priori probability. The 

 extent to which we are able to make use of this function, then, depends 

 on how precisely we are able to assign numerical values to it before 

 we study our sample. 



1 This general type of problem has been under study within the Bell System for 

 some time. In an article "Deviation of Random Samples from Average Conditions 

 and Significance to Traffic Men" by E. C. Molina and R. P. Crowell which ajipeared 

 in the Bell System Technical Journal for January, 1924, a special case of sampling 

 theory was developed and various possible applications were suggested. In August, 

 1924, Molina delivered a paper entitled "A Fornmla for the Solution of Some Prob- 

 lems in Sampling" before the statistical section of tiie International Mathematical 

 Congress in Toronto, Canada. This paper dealt with a somewhat more general 

 case of the sampling problem than was discussed in the article just mentioned. 



