QUALITY CONTROL CHARTS 601 



term Gram-Charlier series or Pearson IV type in this case) would give 

 as large or larger value of x" than that observed. Therefore the basis 

 for the conclusion at the end of the previous paragraph is that we have 

 faith^ that the customary method of taking theoretical steps 1 and 2 

 gives a close approximation to the true distribution of the product 

 when it is uniform or controlled. 



Turning to a study of the data grouped into monthly distributions 

 (Fig. 2), we find additional evidence of lack of control. Naturally the 

 monthly obser\'ed \-alues of the four statistics, average X, standard de- 

 \-iation a, skewness k = \/^i, and kurtosis /So should lie within well- 

 defined limits established by sampling theory (step 3) and shown in 

 Fig. 4, if the product had been controlled. Furthermore, the obser\ed 

 values of percentage defective p (percentage of instruments ha\ing 

 quality less than some value X) from month to month also should fall 

 within well-defined limits. Using the grand average^ of a statistic as 

 the basis for establishing limits, the first fi\'e sections of the control 

 chart in Fig. 4 were constructed. The dotted lines calculated upon the 

 basis of a uniform sample of 1250 indicate the limits within which the 

 different statistics should lie, if the product had been controlled. The 

 chart shows that observed values of these statistics often fall outside 

 their respective limits indicating, subject to limitations imposed by the 

 method of calculation, lack of control of product. 



We may go still further and, without carrying out the analysis of 

 Fig. 3, make use of Pearson's test of goodness of fit to calculate the 

 probability that the first two months' samples could have been drawn 

 from the same universe (the same uniform product), then that the 

 third month's sample could have come from the same universe as the 

 combined samples for the first and second months, etc.^ Obviously the 

 values of x^ used as a basis for this calculation of the goodness of fit 



' Such faith may be based upon the a priori conception that an observed difference 

 in two values of X is the resultant effect of a large number of causes (following in the 

 steps of Laplace, Charlier, Edgeworth, Gram, Thiele and others^ and upon the ex- 

 perience that observed homogeneous distributions always have been fitted by some 

 one of the well-known forms of probability curves (following in the steps of Pearson 

 and others). 



* Some objection may be raised to the use of the observed average as a basis for 

 establishing the limits of a given statistic, because this observed average almost cer- 

 tainly would not be the true value even though the product had been uniform. In 

 the present case, however, we are probably justified in using the observed average 

 because previous experience based upon thousands of observations has given approx- 

 imately the same values for these quantities. Rigorously, of course, we should find 

 the standard deviations of monthly differences from the grand average and set up 

 limits on this basis. Wherever necessary this method is followed and in fact has been 

 carried out for the case in hand where it gives results similar to those indicated in 

 Fig. 4. 



9 Pearson, K.P., Biometrika, vol. viii, 1911, p. 250 and vol. x, 1914, p. 85. 

 Rhodes, E.G., Biometrika, vol. xvi, 1924, p. 239. 



