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BELL SYSTEM TECHNICAL JOURNAL 



if the same individual had fired at both targets. In other words, non- 

 assignable, fortuitous or chance causes introduce certain differences in 

 the average, dispersion and shape of the observed polygons from one 

 month to another, and we must set up some method of differentiating 

 the effects of assignable from those of non-assignable causes. 



Outline of Basis for Detecting Lack of Control 



Uniform product was defined above as one for which the differences 

 between the units or groups of units were controlled by a complex 

 system of non-assignable chance causes producing results independent 

 of time. Now, following a line of reasoning whose origin is attributed 

 to Laplace, it may be shown that such a system of causes, in general, 

 may be expected to give a unimodal distribution of product such that 

 the probability dy^^- of the production of a unit having the quality X 

 within the range X to X+dX is independent of time, being a contin- 

 uous function, /', of the quality X and certain parameters. We may 

 represent the probability symbolically by the following equation 



dy^=f{X, Xi', X./ . . . ym')dX, . (!) 



where the \"s represent the m' parameters. Experimental evidence 

 abounds in many fields of science to justify the adoption of Eq. 1 to 

 represent the probability distribution of the effects of systems of 

 chance causes. It is quite reasonable, therefore, to adopt this equation 

 as a definition of uniform product and to use it as a basis for detecting 

 lack of control. 



Obviously, if we knew /' and the values of the w' parameters in 

 Eq. 1, it would be comparatively easy to determine the limits within 

 which the quality X or any estimate of a parameter derived from a 

 sample of the product might be expected to vary because of chance 

 causes. In practice, however, we know only the n observed values of 

 quality obtained from inspecting a sample of as many units, and we 

 do not know either the true functional relationship/' or any one of the 

 m' parameters even though the product be uniform. We wish to find/' 

 and each of the w' parameters, but, knowing that we cannot do this, we 

 try to find some approximation / for the true function /' and some 

 estimates 0i, 02 • • • 9m for the parameters Xi, X2 • • . Xm occurring in 

 f. To do this we tentatively assume that the sample of n units has 

 been drawn from a uniform product distributed in accord with the 

 function /, and then use statistical theory to see if our assumption is 

 justified. 



