242 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



From these expressions the values of the standard deviations and error correlations 

 are at once found by (viii.). 

 We have 



v ~~ ffl * "*-: ( xv -). 



\/2n ' 

 1 - rf 





E' I- T> ' 1* / \ 



'-.'., = 7. K "-'.. = ^ ....... (XVIII.). 



The result (xv.) is, of course, old ; the results (xvi.) to (xviii.) are novel, and lead 

 to interesting conclusions, which are considered in the following paragraphs. 



(a) The probable error of a coefficient of correlation ?-,, is '67449(1 r^)/^/n. If, 

 therefore, the correlation between two organs is less than once to twice '67449/^/w, 

 they cannot be safely assumed to be correlated at all. 



(ft) If we know definitely the errors made in o-, and a-., if, for example, we know 

 the variations accurately then the probable error of 1\., is that of an array of r t . 2 's 

 for definite ACT, and ACT,. It is given at once by the coefficient of (Ar, 2 ) 2 in (xi.) as 



G7449(l --<'?,)/ x /Ml +/?,)} ....... (six.). 



This is the value given for the probable error of r n by one of the present authors 

 in a former paper.* At that time he had not fully realised the importance of the 

 principle of tbe correlation of errors made in determining the magnitude of 

 frequency constants. 



The following table will enable the reader to appreciate the difference in magnitude 

 between the ' absolute " and " partial " probable errors of r, 2 : 



1 1 



1 -99 -985 



2 -96 -94 



3 -91 -87 



4 -84 -78 



5 -75 -67 



G -64 -55 



7 -51 -42 



8 -36 -28 



9 -19 -14 



1 



"' Heredity, Regression, and Panmixia," ' Phil. Trans.,' A, vol. 187, p. 266. 



