RELATION OF VARIABLES. 129 



This is not an easy question to answer. However, the validity of 

 the second assumption obviously depends upon distribution of the 

 values of the dependent variable. If they are distributed in approxi- 

 mate accordance with the Gaussian, or normal law of error, the second 

 assumption may be regarded as true, while, if their distribution devi- 

 ates widely from this law, it is clear that this assumption is not valid. 

 In such cases it is desirable to find some function of the dependent 

 variable that is distributed in accordance with the Gaussian law, and 

 deal with that function. For example, in many, if not most, biologi- 

 cal problems, the change in w is not independent of the magnitude of w, 

 but approximately proportional to it, and the logarithm of tv, rather 

 than w itself, is distributed in accordance with the Gaussian law. 

 Accordingly, the proper form of expression, upon which to base 

 the mathematical reasoning, becomes (w -\- k) = /i(.r) XMy) Xfsiz) 

 etc., where, for greater generality, the constant k is introduced. But, 

 this expression may be written 



log{to + k) = logUx) -\- log My) + logMz) + . . . (131) 



and putting W = log{w + k), Fi = logfi, Fo. = logfo, etc., gives 



W = F^ix) + F^iy) + Fs{z) + . . . (132) 



which is of the same form as that given by equation (1). In general, 

 then, the nature of the frequency distribution of the dependent vari- 

 able, affords an empirical criterion for determining the validity of the 

 second assumption, and for suggesting what function of iv to use to 

 make this assumption valid. 



But this does not wholly eliminate the difficulty. For although 

 the first assumption is, as a rule, valid whenever the second one is, 

 this is not necessarily the case. Furthermore, there seems to be no 

 criterion in the data themselves for determining this. However, in 

 any case, after the "standard" values are selected, the ascertained 

 change in the dependent variable corresponding to a given change in 

 any particular independent one, is the change that would take place 

 under average conditions of the remaining independent variables. 

 It is obvious, then, that, when the range in value of the independent 

 variables is small, the error introduced by this assumption is corre- 

 spondingly small. Therefore, by classifying the data so as to restrict 

 the range of those independent variables with respect to which the 

 assumption does not accord well with fact, and applying the method 

 of successive approximation separately to each portion of the data 



