1892.] Correlated Variations in Crangon vulgaris. 3 



connected that when the deviation of one variable from its average is 

 known the mean deviation of the second is known, then, evidently, a 

 surface can, from these data, be constructed, showing the relative fre- 

 quency of occurrence of all possible combinations between the two 

 variables. The changes produced in such a surface by changes in 

 the degree of interdependence of the two variables have been in- 

 vestigated, at Mr. Galton's request, by Mr. J. D. H. Dickson (' Roy. 

 Soc. Proc.,' vol. 40, 1886, p. 63). The results of this investigation, 

 which are of importance for the present purpose, are two : — 



(1.) In the population examined, let all those individuals be chosen 

 in which a certain organ, A, differs from its average size by a fixed 

 amount, Y ; then, in these individuals, let the deviations of a second 

 organ, B, from its average be measured. The various individuals 

 will exhibit deviations of B equal to x u x 2 , x 3 , . . . . , whose mean may 

 be called x m . The ratio x m jY will be constant for all values of Y. 



In the same way, suppose those individuals are chosen in which 

 the organ B has a constant deviation, X ; then, in these individuals, 

 y m , the mean deviation of the organ A, will have the same ratio to X, 

 whatever may be the value of X. 



(2.) The ratios x m jY and y,»/X are connected by an interesting 

 relation. Let Q« represent the probable error of distribution of the 

 organ A about its average, and Qj that of the organ B ; then — 



yJX _ 0^ XntlQl, _ WQa _ congfcant 



x m /Y - Qh ' ° r Y/Q« - XI Q b - ^ a COnstant ' 



So that by taking a fixed deviation of either organ, expressed in 

 terms of its probable error, and by expressing the mean associated 

 deviation of the second organ in terms of its probable error, a ratio may 

 be determined, whose value becomes +1 when a change in either 

 organ involves an equal change in the other, and when the two 

 organs are quite independent. This constant, therefore, measures the 

 " degree of correlation " between the two organs. 



A determination of this constant will now be made in the case of 

 five pairs of organs of the shrimp. In accordance with Mr. Galton's 

 notation, the constant will be denoted by r, the mean size of each 

 organ by M, and the probable error of distribution about the mean 

 tyQ. 



The organs measured are shown in the woodcut fig. 1 ; they are : — 



(1.) The total carapace length, measured in a straight line ; 

 (2.) The length of that portion of the carapace which lies behind 



the single gastric spine ; 

 (3.) The length of the sixth abdominal tergum ; 

 (4.) The length of the telson. 



The measurements made were recorded to within 0*05 mm., and 



b 2 



