September S, 1905.] 



SCIENCE. 



309 



mals, man has become the most wonderful 

 living thing on earth, separated by a great 

 gulf from his next of kin, and yet, in spite of 

 his high degree, af&icted with more diseases 

 than any other animal and beset by at least as 

 many tormenting parasites. Inexperienced in 

 his early history, his mind steadily advanced 

 until to-day he contemplates all nature with 

 a yearning to know its mysteries. The 

 changes in the germ-cell sufficing to evolve 

 him are as inscrutable to his reason as the 

 constitution of matter and the interstellar 

 ether, the nature and origin of the cosmical 

 forces and of chemical affinity, the conditions 

 obtaining on other worlds revolving about un- 

 told millions of other suns, or the origin, na- 

 ture and meaning of life itself. But we 

 ardently desire to know these things, to peer 

 out into unfathomable space and to speculate 

 upon the meaning of our existence and the 

 unknowable as we perceive it all about us in 

 the universe. Under such circumstances are 

 we to live but a short time on earth and then 

 be consigned to everlasting oblivion? In con- 

 templating the real significance of the word 

 eternal or everlasting, which must refer to 

 infinity — a duration of time so inconceivable 

 that a number of years expressed in pica type 

 encircling the entire globe would be as naught 

 when compared with it — our reason would ap- 

 pear to answer in the affirmative. But, as a 

 species — sapiens — of the genus Homo, we can 

 never know. We seem to be but intellectual 

 atoms floating in an infinity of space and time. 



Thos. L. Casey. 

 St. Lons, 



August .3, 1905. 



SPECIAL ARTICLES. ' 

 THE SPEARMAN CORREIATIGIs" FORMULA. 



Some time ago C. Spearman published a 

 formula for calculating the true correlation 

 by the Pearson formula for observations in 

 themselves variable.^ This method has been 

 used by several psychologists without a full 

 understanding of the way in which Spearman 

 arrived at the method. Such a use of the 



^ C. Spearman, American Journal of Psychology, 

 Januarv, 1904. 



method is dangerous since those who apply it 

 can not be accurately informed as to the con- 

 ditions under which it holds. 

 Let 



T represent the type. 



0" represent the variability of the group. 



t represent single observations upon individuals. 



!<[ the norm of the individual. 



V represent the variability of the individual 

 with respect to his norm, including the error of 

 observation. 



Assuming that all f s follow the exponential 

 law and representing averages by [ ], we shall 

 find in the long run that 



The observed variability of the group may 

 be expressed as 



in which cr represents the true variability that 

 determines the true correlation in the Pearson 

 formula. The whole problem in practise is to 

 find the value of v. 



Spearmaji says that the true correlation 

 may be obtained by dividing the average corre- 

 lation for the various trials of the two tests 

 by the square root of the product of the corre- 

 lations for the successive trials for each test. 



Let, 



xy = the average product of the deviations for 

 the corresponding single trials in two tests. 



( pg ) J =: the same for t^ and ^2 of the first test. 



(p(jf), = the same for t^ and ^2 of the second 

 test. 



v-y and V.J. = the true variabilities of individuals 

 in the two tests. 



ff^ and Cn = the true variabilities as calculated 

 for X and y. 



Then by substitution in the formula of 

 Spearman, 



l^yY 



Hence, 



(m)i (m)2 



2^= (pq)i ipq)2 



