20 BELI. SYSTEM TECHNICAL JOURNAL 



tions (5) and (8) in terms of the observed \-alues of <je, gq and k.j. 

 In other words we ha\e 



1 -v 



V27r(cr--(r|) 





PART II 



CORRFXTIOK OF DaTA TaKEN PERIODICALLY TO DETECT SIGNIFICANT 



Changes in Quality of Product 



Irrespective of the care taken in defining and controlhng the manu- 

 facturing processes, the units of a product will differ among them- 

 selves in respect to any measurable characteristic. Random fluctua- 

 tions in such factors as humidity, temperature, grade of raw material, 

 and wear and tear on machinery may produce such differences be- 

 tween units of a product. Such random variations in the factors 

 underlying the manufacturing process usually yield a product in 

 which the units differ in random fashion according to some law of 

 probability. 



Customaril}-, product is inspected periodically, and the data are 

 analyzed to determine if the observed difference in two samples is 

 greater than can be accounted for as a random variation. If it is, 

 we may assume that the manufacturing processes have changed 

 significantly for some reason which further investigation should dis- 

 close. Now, the presence of errors of measurement effectively in- 

 creases the magnitude of the random differences to be expected from 

 one sample to another and hence makes it harder for us to detect 

 trends or fluctuations in product. Let us investigate this effect of 

 errors of measurement. 



Symbolic Statement oj Problem 



Symbolically we may assume that the probability of production ot 

 a unit of product having a characteristic X within any range A' to A" 

 ■\-dX is iT{X)dX, where the characteristic X is measured by a method 

 subject to a law of error /£(x), so t\\3.t J E{x)dx represents the proba- 

 bility of occurrence of an error x within the range x to .v+(/.v. The 

 problem is to find the corresponding distribution /,.(A') for the ob- 

 served magnitudes. 



