LETTERS TO THE EDITOR 179 



to indicate to the casual reader that there are certain technical questions 



involved in its application which must be given due consideration even 



beyond the stage outlined in Mr. Fisher's communication. I have 



found that this series often has many advantages over competing 



methods of analyzing data although not all of these advantages are 



referred to in the literature of the subject. 



W. A. Shewhart. 

 December 28, 1926. 



To the Editor of the Bell System Technical Journal: 



The question raised by Dr. Shewhart as to the measure of the 

 absolute magnitude of the difference between a known function, 

 F{x), and the first n terms of the series has been treated by Gram in his 

 original article on " Rcekkeudviklinger hestemte ved Hjcelp af de mindste 

 Kvadraters Metode." (On Development of Series by means of the 

 Method of Least Squares.) In this article Gram also discusses at 

 length the decidedly practical question of arriving at an estimate of 

 the remainders (or residuary terms), which invariably occur in practice 

 where we, of course, are forced to deal with a finite number of terms. 



It would, however, be beyond the limits of the present communi- 

 cation to enter into this aspect of the question, which necessarily is 

 somewhat complicated. In passing it, I wish merely to state that 

 Gram's original method of determining the coefficients in the series 

 on the basis of the principle of least squares is decidedly easier to apply 

 than the relatively cumbersome method of moments in arriving at a 

 reliable measure of the remainder of the series after, say, the w*"^ term. 



Dr. Shewhart's further contention that two terms of the Gram 

 series sometimes give as good a fit as three or even four terms, and that 

 three terms in the case of nearly symmetrical distributions serves as 

 well as four terms, seems to me to be almost self-evident from a simple 

 consideration of the way in which the coefficients c actually enter into 

 the series. 



All the terms containing uneven indices tend to produce skewness, 

 and all the terms with even indices produce excess (kurtosis). If the 

 coefficient Cs is not too large, and if r4 is small as compared with Cs, it is 

 evident that 



F{x) = ro^o(s) + C3<p3(s) 



will give about as good an approximation as 



F(x) = Coipoiz) + C3^.3(s) + CiiPi{z) + Ics-ifeiz). 



On the other hand, in nearly symmetrical distributions with a pro- 



