728 BELL SYSTEM TECHNICAL JOURNAL 



markable, even though this be true, that the work of Tchebycheff ^ 

 makes it possible for us to give a lower bound to the probability that a 

 unit of_product will be produced with a quality X lying within the 

 range X' ± Li and also therefore to the probability that an observed 

 average quality of a sample of n units will lie within any given range 

 X' ± Ln. 



Taking Li = ccr' (c > 1), the probability Pc<t' that the constant 

 system of causes Type III will produce a unit of product having a 

 quality X within the range X' ± Li is given by the expression 



^-'^1-^- (2) 



Expression 2 also defines the probability that the average quality of n 

 units of product coming from the constant system of causes Type III 

 will lie within the range X' ± L„ where 



T _ '^^' 

 ■Ln — ~7=* 



■\n 



Let us illustrate the method of finding the limits under these specifi- 

 cations. Assume that the specified average resistance X' of a relay is 

 150 ohms and the standard deviation <j' is 5 ohms. What is the range 

 within which we may expect 90 per cent of the product (i.e. Par' = .90) 

 to lie, assuming no assignable causes of variation in product? What is 

 the similar range for the average of 1000 relay windings? 



Turning to the nomogram of Fig. 4, we connect by a straight line the 



point Pea' = -90 and the point A near the center of the chart. The 



point on the c scale fixed by the intersection of the straight line so 



determined with the c scale is 3.15. The required values of Li and 



3 15 X 5 

 Liooo are therefore Li = 3.15 X 5 = 15.75 ohms and Liooo = 



-VlOOO 

 = .50 ohm. Hence the limits are 150 ± 15.75 ohms and 150 ± .50 

 ohms. 



To avoid the slide rule computations in obtaining ca' and ca'f-yjnwe 

 can use the nomogram of Fig. 5. We enter this nomogram by the 

 value c = 3.15 and find a point on the c/Vw scale which lies on a straight 

 line with the point c = 3.15 on the c scale and w = 1 on the n scale. 

 Connecting the point thus fixed on the c/^Jn scale with the point 

 a' = 5, we read on the L scale 15.75 ohms. Carrying through the 

 same procedure, but starting with n = 1000 instead of w = 1, we read 

 on the L scale .50. These values give the limits found above. 



* Tchebycheff, Liouville Journal, 1867. " Des valeurs moyennes," Journal de 

 Mathematiques (2), Vol. 12, pp. 177-84. 



