244 PROFESSOR K. PEARSON AND MR. L. N. G. FILQN 



while a selection of variability may produce only a small or moderate change on the 

 variability of correlated organs, a selection of correlation or a selection of variability 

 is likely to produce considerable changes on variability and correlation respectively. 

 Let tr 1; cr 2 , r n , be the mean values of the standard deviations and the coefficient of 

 correlation for any three organs ; let cr, -f- ACT,, o- 2 -f* Ao- 2 , r, 2 + A>' J2 , be the like 

 quantities for a group selected at random. Then the principle of regression tells us 

 that most probably 



Substituting the values given by (xv.) to (xviii.), we find 



''a i 



Ao-, .= 0-, 1 ---- ; Ar,, 



. 



' 



Now these equations lend us to some important conclusions. In the first place, if 

 the correlation be very small or very large, then a random selection of variability (Ao-,) 

 makes only a small change in correlation (Ar, 2 ). The change in correlation for a 

 selection of variability is greatest when r t ., l/\/3, and then is approximately 

 385 Ao-i/cr,, or over 6 per cent., if Ao-,/0-, were as high as -^j. On the other hand, the 

 change in variability (Ao-,) due to a selection (A'/ 1 ,.,) of correlation is small if the 

 correlation be small, but increases rapidly if the correlation become nearly perfect. 

 Of course, for perfect correlation the probable error of r l2 is zero, and accordingly it 

 is infinitely improbable that a selection can be made with Ar, 2 differing from zero. 

 But if r v , be not unity, then a selection in which Ar 12 is large, however improbable, 

 will give very large changes in the variability, if r, 2 be very large. Our conclusion is 

 accordingly that considerable changes in variability are likely to be produced whenever 

 there is a correlation selection among highly correlated organs. 



(7) To find the Probable Error of the Regression Coefficient for Two Organs. 

 The regression coefficient />, is given by 







Pi = r n a-i/cr z . 

 Its standard deviation S P| is given by the summation equation 



