SHORTER ARTICLES AND DISCUSSION 



A NOTE ON CERTAIN BIOMETFJCAL COMPUTATIONS 1 



1. It is a. well known fact that curves of individual growth, in 

 which some size character of the organism is taken as ordinate, 

 and time as abscissa, are closely related to a logarithmic curve. 

 To Pearson 2 belongs the credit of first demonstrating this con- 

 cretely by fitting a logarithmic curve to growth data. Since that 

 time a number of other students 3 of growth have made use of 

 such curves in <:raduatin<: observat ional data. 



Now while the simplest logarithmic curve 



y = a+b log x (i) 

 is probably only very exceptionally (if ever) followed precisely 

 in the growth of an organism, yet it certainly represents the 

 general type towards which many observational growth curves 

 tend. In the practical analysis of growth data it is often found 

 to be extremely helpful as the first step to fit such a curve as (i) 

 or a simple variant of it in which a "line" term is added, as in 

 y = a -f! bx + c log x. (ii) 

 Actually finding out by trial just wherein a curve like (ii) fails 

 to fit the data— if it does fail — will usually give one the clue as 

 to the way in which the curve must be modified in order to grad- 

 uate the observations satisfactorily. 



In fitting a curve like (ii) to a series of observations by the 

 method of least squares the type equations are as follows : 

 S(y)—na — bS(x)-cS(logx)=0, . 1 



S(xy) —aS(x) —bS(x*) —cS(x log x) =0, Kiii) 

 8(y log x) — aS(\og x) — bS(x log x) — c8(log x) 2 = 0, J 

 where S denotes summation for the n values of the variables. 



Now it is evident that, of the 11 summations included in these 

 equations, only 3 involve the variable y. All the others are func- 



1 Papers from the Biological Laboratory of the Maine Agricultural 

 Experiment Station. No. 31. 



2 Pearson, K., Biometrika, IV, 131-190. Cf. also Lewenz, M. A., and 

 Pearson, K., ibid., Ill, 367-397. 



3 Cf., for example, Pearl, R., Pepper, O. M., and Hagle, F. J., Carnegie 

 Institution Publ. No. 58, 1907, and Donaldson, H. H., in Jour. Comp. Neurol, 

 and Psychol., XVIII, 345-392, 1908, and also in later papers. 



756 



