86 BELL SYSTEM TECHNICAL JOURNAL 



observed standard de\iation di\ided In- tlu- sciiiarc root of the luiiuber 

 of observations, in this case 10,000. 



VVe have already seen, however, that it is possible to detect such 

 errors of sampling, since in general the distribution cannot be fitted 

 by the second approximation or Gram-Charlier series. If the theo- 

 retical distribution is either normal, second approximation, or the 

 law of small numbers, the number of observations to be expected 

 between any two limits can be readily determined from the tables. 

 Experience has shown that in e\ery instance where it has been possible 

 to represent the observed distribution in any of these three ways, the 

 data obtaineii in future samplings iiave alwa\s been consistent with 

 the results to l)e expected from the lheor\- imderlying these three laws. 

 It will be of interest to note the data gi\en in columns 3, 4, and 5 of 

 Table X'lII and to compare the theoretical percentages (last row) for 

 the different limits with those obser\c<l. 



In closing it is of interest to point out further liie significance of 

 some of the results discussed in this paper in connection with the 

 inspection of equipment. Here we must decide upon a magnitude 

 of the sample to be measured in order to determine the true percentage 

 of defectiNe instruments in the product. If p is the percentage 

 defective, and q that not defective, then the standartl de\iation about 

 the average nimiber found in a sample of n chosen from -V instruments 



'=^^"(^-:v)- 



In practici', ImwA-xfr, we ne\er know tin- inu- x.ilui- ol (> unless we 

 measure all of the apparatus, and this is inipraciiial. In our calcula- 

 tions we must therefore use some corrected value. We find, though, 

 that the average value of p is in most instances the one that must be 

 used. Assuming that we choose a value of /;. ihr distribution of 

 defectives in A'' samples of n in number will he represented b\' the 

 distribution of X'{p-\-</)". If cnu- of the samples is found lo contain 

 a percentage of defeclixes, which is inconsistent, that is, which is 

 highly imjirobable as iletermined from the distribution of X'(p+q)", 

 it indicates that the product is changing. 



If, however, we take mto account the efleci of the size of the first 

 sample in respect to the second as indicated 1)\- Pearson," we see that 

 the distribution of A" s;im])les may be dilTerenl from ilial given 1)\- 

 the binomial expansion. In accordance with this ilu'or\-, if in a tirsi 

 sample of 100, 10' i of the sani|)le i> found lo possess a gi\en al iriliute, 



*• Pearson, K. Luc. cit. 1-uut note 31). 



