QUALITY CONTROL CHARTS 599 



Thus, if the n observed values of X are grouped into m + 1 cells 

 having frequencies no, Wi, . . . w,„-).i and if the calculated or theo- 

 retical frequencies in these same cells as determined from Eq. 3 

 are noQ, Wiq, . . . «,„o where '^ni = 'En,Q = n, we may calculate by 

 Pearson's method the probability P of random samples exhibiting 

 as large or larger values of X" than that observed in our sample 



where ^2 = -^ ^^'Q~'^'^\ If the value of probability P thus found 



fliQ 



is small, we may conclude that it is highly improbable that 

 the sample of n units of product came from uniform product of the 

 form assumed. Of course, this theoretically does not settle the 

 question as to whether the sample might have come from a uni- 

 form product other than that assumed, because, as we see, /is only 

 an assumed form for/'. Practically, however, we seem justified in 

 concluding that it is unlikely that the product is uniform if P is 

 small, particularly since the choice of/ is customarily made upon 

 the basis of large samples. The application of this test is illus- 

 trated in connection with the discussion of the data in Fig. 3. 



Practical Application of Theory 



The application of the steps just outlined will be illustrated by an 

 analysis of the data in Figs. 1 and 2 to show that the product had 

 not been controlled for the period therein indicated. Carrying out 

 steps 1 and 2 we conclude that the best theoretical equation represent- 

 ing the data in Fig. 1 is either^ the Gram-Charlier series (two terms) or 

 the Pearson curve of type IV for both of which the estimates of the 

 parameters may be expressed in terms of the first four moments ni, /X2, 

 fjLz and m of Fig. 3. These two distributions are shown in columns 10 

 and 14 respectively.-^ Pearson's test for goodness of fit (step 4) gives 

 negligible results^ (the probabilities of fit as measured by P on the 

 chart are for practical purposes zero) in both instances, and this was 

 taken as indicating that assignable causes of variation had entered the 

 product. Further investigation of an engineering nature justified this 

 conclusion. 



We should not fail to note as suggested above, however, that a small 

 value of fit technically indicates only that the chance is small that a 

 random sample drawn from the theoretical universe (either the two- 



'' Equations for these curves may be found in Bowley's Elements of Statistics, 

 pages 267 and 345 respectively. 



^ Bowley's table, page 303 in his "Elements of Statistics," was used in the calcula- 

 tion of the Gram-Charlier graduation. 



® Corrections were applied to take account of the number of degrees of freedom, 

 etc., in the calculation of goodness of fit. 



