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BELL SYSTEM TECHNICAL JOURNAL 



has been found satisfactory in many cases to take P =• .99 and so, 

 upon this basis, we shall present the method of setting limits upon the 

 expected fraction defective in a sample of size n. It is well known that 

 the probability of an observed value of p lying within the limits 

 p' zb Scp, is approximately equal to .997 provided the fraction de- 

 fective p' is approximately equal to the fraction non-defective q' , and 

 n is large. It can be shown, however, that irrespective of the magni- 

 tudes of p' and n the value of P so determined lies between .95 and 1.00 

 and for most cases met in practice P does not differ from .997 by as 

 much as 1 per cent. 



It is obvious, therefore, that, if we construct an alignment chart on 

 which we may read directly the standard deviation a^, when p' and n 

 are given, then the average p' plus or minus three times the standard 

 deviation a^, gives the corresponding values of the limits. 



Let us consider a practical problem, see how the question of whether 

 or not a product is controlled really arises and see how control limits 

 can be found from the alignment chart of Fig. 1 to answer this question. 



Table 1 represents the observed fraction found defective over a 

 period of 12 months for two kinds of product designated here as Type 

 A and Type B. The table gives for each month the sample size n, 

 the number defective ni and the fraction defective p = mln. The 

 average fractions defective for the 12-month period are pA = .0109 

 and pB = .0095. Subject to later consideration we shall assume 

 Pa = Pa and ps = pB. 



TABLE 1 



