QUALITY CONTROL CHARTS 597 



Theoretically there are four fundamental steps in the procedure out- 

 lined above. They are: 



1. The Problem of Specification: To find or specify a satisfactory 

 form /of the distribution of the uniform product from which the 

 sample of n pieces is assumed to have been drawn or to find the 

 equation 



d\\ = f{X,\,, Xo, . . . \,)dX (2) 



where dy^^ is the assumed probability of a unit having a quality X 

 within the interval X to X-\-dX. 



For example we often assume the distribution to be normal so 

 that Eq. 2 becomes 



1 _(X-m^)_2 



Here m = 2, and Xi and X. are respectively the arithmetic mean nn 

 and the root mean square (or standard) deviation a of X as defined 

 by the normal curve Eq. 2'. 



2. The Problem of Estimation: To find from the data given by 

 the sample a suitable estimate for each of the m parameters in 

 Eq. 2. These estimates of the parameters in terms of the data of 

 the sample are often termed statistics. If we let 0, represent the 

 chosen statistic for the parameter X, in Eq. 2, we may rewrite this 

 equation as follows 



dy^=f{X, 01, Go, . . .Qm)dX (3) 



as our theoretical approximation for the assumed true (Eq. 2) 

 probability distribution. 



An estimate of a given parameter may often be obtained in a 

 number of ways by one or more methods. 



In the above illustrative case of the normal law, we must esti- 

 mate the two parameters nii and a (Eq. 2') from the n observed 

 values of X in the sample. Now, it is well known^ that a may be 

 expressed in an indefinitely large number of ways in terms of the 

 arithmetic means of the absolute values of the integral powers of 

 the deviations of X defined by Eq. 2'. Estimates of a might be 

 obtained in terms of the corresponding means calculated from the 

 2 Whittaker and Robinson, Calculus of Observation, page 182. 



