174 BELL SYSTEM TECHNICAL JOURNAL 



which the possible combinations of the "elementary errors" actually 

 enter into the expansion. It can be shown that there exists a definite 

 relationship between on the one hand the relative order of magnitude 

 of the elementary errors and, on the other, the arrangement of the 

 individual terms of the Gram series.- 



This relationship was already established by Thiele. It was prob- 

 ably first concisely formulated by Edgeworth, and later on by Charlier 

 and Jorgensen. 



The various degrees of approximations can be expressed by the 

 following schemata : 



1st approximation (/7o(z), 

 2d approximation (Pq{z) + £3(^3(2), 

 3d approximation <^o(s) + Cz(p?,{z) + Ciipi{z) + C(,(P(,{.z), 

 4th approximation (^0(2) + c^ipziz) + Ci^pAiz) + C(,<p^{z) + ^5(^5(2) 

 + c^^p^{z) + ^9(^9(2). 



The first approximation is the usual normal curve. The second is 

 the one which the English statistician, Bowley, erroneously thinks 

 represents a generalized frequency function and for which Dr. Shewhart 

 has shown a marked preference. The third approximation, except for 

 the term involving the sixth derivative, has been used very extensively 

 by Charlier. 



Through the publication by C. V. L. Charlier in 1906 of extensive 

 tables to four decimal places of the third and fourth derivatives, the 

 Gram series was made available for practical statistical work in the case 

 of frequency distributions with a moderate degree of skewness and 

 excess (kurtosis). But although Charlier was aware of the fact that 

 the retention of the fourth derivative- — which is related to excess 

 (kurtosis) — automatically brings about the inclusion of the sixth 

 derivative, it was not before Jorgensen issued his large numerical 

 tables of the first six derivatives to seven decimal places that we were 

 able to do full justice to the third approximation of the Gram series. 

 Incidentally it might in this connection be mentioned that it is doubtful 

 if the much lauded test for "g(Hxlness of fit" as devised by Pearson 



- Whenever we use the method of moments, the arrangement of the individual 

 terms is not arbitral y but must be made according to "order of magnitude" of the 

 various derivatives; and the orders of magnitudes do not correspond to the indices of 

 the derivatives. The generic term "order of magnitude" has in this instance only 

 reference to the formation of the "elementary errors"; if taken in any other sense it 

 is meaningless. The fourth and sixth derivatives are of the same order of magnitude; 

 while the fifth, seventh and ninth all are of the next order following the fourth and 

 sixth. The concept of the different orders of magnitude of the elementary errors is 

 due to Poisson who already in 1832 arrived at the second approximation of the Gram 

 series. 



