824 



SCIENCE 



[N. S. Vol. XXIX. No. 751 



of ■wliicli according to the usual method of 

 correlation tables occupies much time, while 

 the subdivision into larger groups makes the 

 results inaccurate. The following method of 

 calculation secures a great saving of time and 

 labor. The averages and mean square varia- 

 bilities of all the variables must be deter- 

 mined. By forming the differences between 

 any series of pairs, we find the values of 

 X — y, which may be treated like any variable. 

 Indicating averages by brackets, we have 

 [(a; — 2/)=] = [ar^] + [1/=] — 2[a!2/] 

 = "x" + Cy' — 2rffx(Tj, 

 [(a — y) -] — Cx' — ff/ 



For a single correlation there is not much 

 saving of time in this method of calculation, 

 but in multiple correlations a very large 

 amount of labor is ' saved. 



A similar device may be used in the calcu- 

 lation of correlations of fraternities. When 

 the deviations for members of a fraternity are 

 designated by x^, x„ x^--- x„, 



[{Xi + x, + cc3+ .-. + Xn)°] =n<T-/( 1 -I- n—lr) 



_ l{x, + x,+ ... + x„)n _ J_ 



n{n — i)o--r- 11. — 1 



A similar method will allow the determina- 

 tion of the average correlation of a large series 

 of variabilities. By reducing each variable 

 to multiples of its variability, we find 



m 



+-+■ 



w= 



+:-:)]=«+., 

 m 



+-+■ 





n(n — 1) 



Correlations of phenomena that can not be 

 measured, but only counted, may be treated 

 in the following manner: If two events that 

 have the probabilities p, and Pj are correlated, 

 we may say that those cases in which the 

 event 1 occurs have the probability 1, or a 

 deviation from the normal probability 1 — p. 



Those cases in which the event 1 does not 

 occur have the probability 0, or a deviation 

 from the average probability of — p^ If we 

 call p/ the probability of the event 2 when 

 event 1 occurs, p„" the probability of event 2 

 when event 1 does not occur, and 5, the co- 

 ■ efficient of regression of 2 upon 1, we have 



p,' — p„ = 5j ( 1 — 

 p" — P2 = —q,Pi 



Pi) 



Thus the phenomenon corresponds strictly 

 to that of measurable variables, and the pro- 

 cedure may be followed that is applied in the 

 calculation of the coefficient of correlation of 

 measurable variables. It follows that 



ft; 



P,P2 + <?,p, (1— Pl) 



We designate, as usual. 



3i='-V^ 



Pii^~Pi) 



Pi,:^PiP2-f »• VPi(l — Pi)pAi- 



Pl,2 — PlP= 



p=) 



V>i{l Pl)P2(l P2) 



The correlation between a measurable and 

 an unmeasurable quantity can be determined 

 in a similar manner. When the measurable 

 quantity is grouped as an array of the meas- 

 urable quantity, we find, using the same sym- 

 bols as before, 



[x'J =qi(l — p) 



[x"] = — g,p 



[a;"] = q^ 



[X']- 



"p(l — p) 



. [f']p 



- P) ' 



[x"-\{l — p) 



c Vp(l — p) 



ffs V p ( 1 - 



Prom these formulas, multiple correlations 

 may be calculated according to the same form- 

 ulas as those used for measurable variables. 

 Franz Boas 



Columbia Univeksitt 



the enzymes of ova — influenced by those 



OF SPERM ? 



Some few summers ago, while working in 

 the laboratories of the Biological Station at 

 Woods Hole, Mass., the writer began some 

 experiments to ascertain whether or not the 

 action of the enzymes of ova were in any 

 measure increased or decreased by those of 

 sperm. The problem was suggested by the 

 work of other investigators which showed that 

 some enzymes have an interdependent action. 

 It was also conceived that the process of fer- 

 tilization might be due to the acceleration of 



