QUALITY CONTROL 727 



Taking n equal to the average sample size (483 for Type ^), we con- 

 nect by a straight line the point 483 on the n scale of Fig. 1 with the 

 point .0109 on the p' scale. We read the intersection of this straight 

 line with the o-^,' scale as .0047. Hence the upper limit for p' is 

 .0109 + 3(Tp' = .0250, a value which is exceeded during the months of 

 May and October; and the lower limit is .0109 — ScXp' = — .0032. 

 Of course negative values of p have no significance ; hence we take the 

 lower limit as zero. Following the same procedure for Type B 

 apparatus, we get limits and .0332. 



We see that twice during the year Type A apparatus appears to have 

 been out of control whereas at no time during the year can we say this 



of Type 5.^ 



Now, we shall take up successively the method of finding limits 

 corresponding to specifications involving: 



a. Only one parameter (Type II). 



b. Only two parameters (Type III). 



c. Two parameters and a restriction on the function / over a 



certain range (Type IV). 



d. Four parameters and a specific function/ (Type V). 



We shall find that the limits become progressively smaller in the above 

 order. In fact for Specification Type II no limits can be set and for 

 Specifications Type III and IV the limits are so large as to be in most 

 instances impractical. 



Specification Type II: The expected or average quality shall he X' . 



There is an indefinitely large number of constant systems of causes 

 which would meet this requirement. Associated with each constant 

 system of causes there are specific sampling limits. Sufficient informa- 

 tion, however, is not called for in the Specification Type II to fix 

 sampling limits on the quality of a single unit or on the expected or 

 average quality. 



In other words. Specification Type II is useless insofar as it does not 

 provide for the detection of lack of control in the sense now under 

 discussion. _ 



Specification Type III: The expected or average quality shall be X' 

 and the standard deviation shall be a' . 



Again there is an indefinitely larger number of different cause 

 systems which would satisfy this requirement. However, it is re- 



* Strictly speaking statistical theory only shows that two of the observed deviations 

 in pA are highly improbable upon the assumption that the product had been con- 

 trolled about P'a- It should be noted, of course, that the sample size is not the same 

 from month to month and hence that the limits for a given month should really have 

 been based upon the sample size for that month. However, in the present instance, 

 this method of procedure leads to the same conclusion as given above and hence 

 was not introduced because of necessary complications. 



47 



