QUALITY CONTROL 731 



and c = 3.0 on the c scale by a straight line and thus find a pivot point 

 on the c/V» scale. Holding the ruler on this pivot point, join it suc- 

 cessively with the permanent points of Lj.> and L^^' and with a' taken all 

 on the a' scale and read accordingly Lj', Lp^>, and Lp, on the L scale. 

 After reading Lx', release the pivot point and turn the ruler around 

 the Lx' point so as to join it with the permanent point for L„'. Then 

 read the intersection of the ruler with the inner circular scale L„', here- 

 by obtaining the limit for a'. Thus in five movements of the ruler we 

 find all four limits: ^ 



± Lfc, = ± .23, 



3 ± Lb,' = 3 ± .46, 



± L^, = ± .095, 



1 ± L„. - 1 ± .067. 



Figure 6 presents the graphical representation of the limits thus 

 determined together with limits on y^ assuming that the theoretical 

 frequency distribution was broken up into 13 cells. ^ The irregular 

 lines show the fluctuations in the estimates of these parameters de- 

 termined from four samples of 1000 each drawn under conditions 

 satisfying the specifications just described for X' = {), a' = \, k' = Q 

 and ^2 = 3. Incidentally it should be noted that in every case the 

 observed fluctuations in the estimates of the parameters are well within 

 the sampling limits. This was to be expected because every effort was 

 made in the sampling process to come as close as practicable to the 

 ideal case where no assignable causes of variation were present. In 

 this respect the data of Fig. 6 form an interesting contrast to the data 

 of Fig. 4 of the article referred to in footnote 1, where evidence of lack 

 of control was found. 



Figure 7 makes it possible for us to set limits about the average or 

 expected x^ corresponding to a probability of either .98 or .80. Thus 

 for the data of Fig. 6 the limits for %" corresponding to probability .98 

 are approximately 3 and 26 respectively as read from this chart. If 

 limits corresponding to any other probability are desired, they can be 

 readily obtained from tables for goodness of fit.^° 



We are now in a position to consider more in detail the advantages 



* In case the given data bring the readings on the extreme points of the scale 

 (where a' > 10) it is advisable to take o-'/lO and multiply the final results obtained by 

 ten. It is also helpful to remember that the L-scale on the nomogram of Fig. 5 can 

 be considered as a regular scale of the product of two factors read on a' scale and 

 cj-^n scale. 



' For the significance of x^ as here used, see paper, footnote 1. 



^^ Elderton's Tables for Goodness of Fit reproduced in Pearson's "Tables for 

 Statisticians and Biometricians" and also R. A. Fisher's "Tables for Goodness of 

 Fit" given in his recent book "Statistics for Research Workers" will be found very 

 helpful in the construction of curves similar to those of Fig. 7. 



