58 fiELL SYSTEM TECHNICAL JOURNAL 



If tin- (listril)utiiin in X is normal such that wi- ha\i> 



1 (^-°)' 



yi = ;=e 2.72 . 



then 



I \,j>(n)-aY 



v = 



<^V2^ 



(«)e 2a^ (4) 



where a is the arithmetic mean value, therefore, the distribution 

 of the causes need not be normal; conversely if the causes are dis- 

 tributed normalh', the observations will not in general be normal. 



This idea is of great importance in the interpretation of observed 

 distributions of physical data.'' To illustrate, let us assume that the 

 natural causes which affect the growth of apples on a given tree 

 produce a normal variation in the diameters of the apples. Ob\iously, 

 the distribution of either the cross-sectional areas or the volumes 

 will not be normal. '^ If the distribution of the diameters is normal 

 as supposed, the arithmetic means of these diameters is the most 

 probable value. Obviously, however, nritlu-r the arithmetic mean 

 area nor the arithmetic mean xolunic will \>v the most probable, 

 because in general 



II )l ' 



As already indicated, tiie de\'ialions tk'alt wilii in liu- present in\esti- 

 gation were not small. The form of the 1 b-crxed distribution ma\' 

 be expected, therefore, to depend upon ilu- fimctional relationship 

 between the obser\-ed (juantitN- and the niunber of causes. We shall 



" Kapteyn, J. C. — Skew Fiiqiiency Cur\is — I ironiiiiicn, 1903. 



" In the theory of errors this fact is taken into acconnt hy assuming that 

 the variations are always siiuill. Thus, if the variahle .V can be represented as 

 a fuiiclion /•' of certain other variables t'l, (':, ... I'm so tliat we have 



-V =/■((•„ r'„ ... i'm), 



we ordinarily assume that we can write this expression in the I'nllowin); form 



A' =i-'(<ii -f "1. Os + Hi, ... a„+Un). 



A further assumption is made that the u's arc small so that 2nd and higher 

 powers and products of these can be neglected. L'ndcr these conditions the 

 distribution of A' is normal and has a standard deviation given by the following' 

 expression : 



But, thus, we arc led to overlook the significance of the form of /■', particularl\' 

 in those ])ractical cases such as are of interest in the present paper where the 

 quantities u,, Mi, ... Km arc not small. 



