734 BELL SYSTEM TECHNICAL JOURNAL 



tions lead us to believe, however, that satisfactory limits can be estab- 

 lished by the method just described making use of the nomogram of 

 Fig. 5 provided the following restrictions as to the size of the sample 

 are made. 



{a) The expected distribution of the averages of samples of any size 

 n is normal about the expected value X' . 



(b) Comparatively small error " will be made in fixing the limits on 

 the parameter a' by means of the nomogram of Fig. 5 provided n is 

 25 or more. 



(c) For a sample of size n of 500 or more the nomogram of Fig. 5 

 may be used in fixing limits on all four parameters. ^^ 



These limitations do not require necessarily that the distribution of 

 the estimate of a parameter must be normal for n as large or larger 

 than specified; instead they merely require that it may be represented 

 by the first few terms of the Gram Charlier series for which the normal 

 law integral over a range equally divided by the expected value of the 

 parameter is a close approximation to the integral of the Gram Charlier 

 series over the same range. 



Fixing the Parameters 



There are various ways of arriving at the values of the parameters 

 to be accepted as the basis for quality control. Sometimes they may be 

 fixed by the economics of the problem. Such is the case for the Type 

 I specification when the economic standard fraction defective or p' is 

 known. At other times the parameters are fixed by technical con- 

 siderations such for example as in the case of an induction coil whose 

 inductance must lie within well-defined limits in order to obtain a 

 proper functioning of the entire circuit, for this would effectively fix 

 X' and a'. In most practical instances the technical considerations 

 tend to fix only the average and standard deviation. At other times 

 we may empirically choose the observed estimates of these parameters 

 determined from the data obtained within the fixed interval of time 

 wherein we have reason to believe the quality has been produced under 

 essentially the same conditions. Irrespective, however, of what 

 period is chosen as a base in fixing p' or any other parameter, the control 

 chart serves to show whether or not the product has been controlled 

 over this period. In any case the parameters are accepted at least as 



" Pearson, Karl, "On the Distribution of Standard Deviations of Small Samples," 

 Biometrika, Vol. X, Part IV, May 1915, pp. 522-529. 



'2 Pearson, Karl, and others, "On the Probable Errors of Frequency Constants," 

 Biometrika, Vol. XIV, 1903, p. 273 seq.. Vol. IX, 1913, p. 22 seq. Isserlis, L., "On 

 the Conditions under Which the Probable Errors of Frequency Distributions Have a 

 Real Significance," Proceedings of the Royal Society, Series A, Vol. XCII, 1915, pp. 

 23-41. 



