36 EMPIRICAL STUDIES OF MEASUREMENT 



ry^ the mean correlation between the various grad- 



ings for p and those for q ; 

 and r p ,, g ,, = the correlation of the amalgamated series for p 



with the amalgamated series for q. 



He has been criticized with some venom bv Karl Pearson 

 (Biometrika, Vol. III., p. 160), who believes these formulas wrong, 

 and concludes that "Perhaps the best thing at present would be 

 for Mr. Spearman to write a paper giving algebraical proofs of all 

 the formulas he has used, and if he did not discover their erroneous 

 character in the process, he would at least provide tangible material 

 for definite criticism, which it is difficult to apply to mere unproven 

 assertions. ' ' 



These formulas of "Spearman's, if correct, are of importance. 

 They should be proved valid or replaced by formulas that are valid. 

 The first formula may be replaced by 



* ovp, 2 1/oy 2 v tq ? 



where r pq and r p > q > are as above and 



%/ = the mean square deviation of the series of measures of p ; 



oy = the mean square deviation of the series of measures of q ; 



o-q,, = the mean square deviation of the different measures of p in 



the same individuals ; 



ov = the mean square deviation of the different measures of q in 

 the same indivduals. 1 



The presupposition of this formula and of Spearman's first 

 formula is. that the attenuation is due to chance errors. Dr. Clark 

 Wissler has called attention to the fact that, where practise, fatigue 

 and other constant influences help to cause the different observations 

 of a fact to vary, these formulas will, therefore, pive inaccurate 

 results. 2 ^) ^ / 3 > 



Of these two formulas, Spearman's possesses the advantage of 

 being usable in cases ^Fere the twcTtraits are not measured in units 

 f 5 ) o amount, such as allow the variabilities of the two traits to be 

 calculated; the formula of Boas has the advantage of being more, 

 rapid and convenient in cases where the variabilities of the two 

 traits can be calculated. 



No active attention has so far as the writer knows been yet given 

 ^ to formula (2) above. 3 Practical necessity seems to justify the labor 



1 This formula is due to Professor Franz Boas. See also the note by Dr. C. 

 Wissler in Science, Vol. XXII., p. 309 ff. 



2 Loc. cit. in note 1. 



* Spearman's second formula has the advantage of measuring the probable 

 true correlation by the actual changes produced in the obtained ' raw ' correlation 

 by a certain increase in accuracy. The nature and validity of the presupposi- 

 tions upon which it is based I am not competent to discuss. 



