THE L3 SYSTEM — QUALITY CONTROL REQUIREMENTS 957 



or small groups of units do not follow one another through the process 

 in time sequence. Here ''order of production" has no meaning for in- 

 dividual units — the only order is the sequence of successive batches. 



For this situation a batch method is provided whereby a substantial 

 sample, normally 50 units, is selected at random from each batch. The 

 average, X, and the standard deviation, a, of the sample are computed 

 and the satisfactoriness of the distribution of the batch is determined by 

 comparing (1) the sample average, and (2) the sample standard devia- 

 tion with certain limits of allowable variation which have been estab- 

 lished. As in the case of the control chart method, consideration is given 

 to the results to be expected in samples from the two acceptable dis- 

 tributions already defined. 



The A50 limits, the limits to be met by a sample average, are 3-sigma 

 control limits for averages on either side of the process average band 

 given hyN zk {V^a" -f SV^^/VSO), which after substituting a" = 0.3^, 

 gives 



A50 limits = A^ db 0.23A. 



The limit to be met by a sample standard deviation is the upper 

 3-sigma limit of a sampling distribution of standard deviations for sam- 

 ples of 50 from a universe having a standard deviation, o-''. However, 

 in practice the standard deviation is difficult to calculate. In order to 

 simplify calculations, an estimated standard deviation is used instead, 

 computed as follows: Divide the 50 values into random subgroups of 5; 

 find the range, R, of each of the subgroups; compute the average range, 

 R, and multiply by 0.43. Thus a (estimated) = 0.43jR. Considering the 

 sampling distribution of this statistic (0.43 times the average range for 

 10 samples of 5) as a substitute for o-, we make use of known theoretical 

 relations between the distribution of R for samples of 5 and the standard 

 deviation of the sampled universe. The upper 3-sigma limit of the 

 sampling distribution of R (for 10 samples of 5) is given* by 



2.3260-" + 3 



/a864^\ 



V VToj 



which, after substituting a'' = 0.3 A, gives 0.95^1. From this the limit 

 to be met by the sample standard deviation (estimated as 0.435) is 



S50 limit = 0.41^1, max. 



The limits for the sample average must be met by each batch, but 



A.S.T.M. Manual on Quality Control of Materials, Am. Soc. for Test. Mat., 

 Phila., 1951; see related formula D2 , p. 114, and factors d2 and da , p. 115. 



