598 BELL SYSTEM TECHNICAL JOURNAL 



sample. Two such estimates familiar to all are (letting 62 stand in 

 general for an estimate of a, the second parameter of equation 2') 



'■'=\2 



^\X-X_ 



and 



922- \^ 



where the summation extends over all the X's in the sample of n 

 and X is the arithmetic mean of these values of X. 



Thus for every X occurring in Eq. 2, we may have many ways of 

 securing an estimate from the sample. Of these ways, which one 

 shall we choose? Obviously, as in the case of 621, compared with 

 622, one estimate may require less labor than another in its cal- 

 culation. This, however, is not always the deciding factor, be- 

 cause one estimate may have a larger error than another. This 

 leads us to the third problem. 



3. The Problem of Distribution: To determine how each of the 

 proposed estimates of a parameter might be distributed in a 

 sequence of samples so that we may obtain some measure of its 

 error. 



In general we desire that estimate of a given parameter which 

 has the smallest error or highest precision. Thus, in the case of 

 621, it requires a sample of 1.14w to give as high a precision as the 

 estimate 622 has for a sample of size n because the ratio of the 

 error of 621 to 622 is Vl-14. Hence the economic savings effected 

 by using the better of two estimates may be very appreciable. 



Furthermore the errors of the statistics are used in establishing 

 the limits within which observed values of the statistics calculated 

 from different samples may be expected to lie as will be illustrated 

 below in discussing the data of Fig. 2. Naturally such errors are 

 used in preparing the control chart Fig. 4. 



Suppose now that we have taken the three steps outlined above 

 and found the calculated or theoretical distribution in the form of 

 Eq. 3. What assurance have we that the observed sample could 

 have come from such a distribution? This question leads us to 

 the fourth problem. 



4. The Problem of Fit: To calculate the probability of fit between 

 the observed and theoretical distributions. 



