240 PROFESSOR K. PEARSON AND MR. L. N. G. PILON 



Further, the standard deviations for errors in h t when the error in h 2 is known, or 

 for errors in h when the error in A, is known, are respectively 



The results (xii.) are, of course, well known ; the results (xiii.) and (xiv.) are, we 

 believe, novel and important. An illustration may be of service. 



Suppose the stature and arm-length of a population to be under consideration. 

 Let us suppose the mean stature of the population known from a great number of 

 observations. Now let the arm-lengths be determined for a random selection of the 

 population ; then, if the stature of these individuals so selected differs A^ in excess 

 from the mean stature of the whole population, the arm-length of the random 

 selection will most probably differ from that of the whole population by A/;..,, a 

 quantity fixed by (xiii.), or rather by the coefficient of regression 



Ro^X/ij/S/ij) i.e., ?'i2cr 2 /crj. 

 Thus 



A/io = r^o-o/o-^A/i,, 



with a probable error of '67449 X ~ (1 r",.,). 



In other words, if the arm-length is to be found from a selection of the general 

 population only, and the stature of this selection differs from that of the general 

 population, it is most reasonable to take the arm-length of the general population to 

 be the mean arm-length of the selected population less the quantity ?-,,cro/cr 1 .A/i 1 . 



Or, again, if a selection from a general population show a mean organ &Ji 1 in excess 

 of that of the general population, the whole system of correlated organs will exhibit 



changes of which the magnitudes are most probably given by the type - -- A/;,. 



The bearing of this on what we have termed random evolution will be obvious. 



(/3) Turning to the second part of the error correlation surface, we note at once 

 that, if two organs be correlated, random selection will give a system of correlated 

 deviations in their variations and their correlation. 



Random selection (and a fortiori it may be added,, artificial or natural selection), 

 which alters an organ's variability, alters the variability and correlation of all other 

 organs. In fact, when it is once realised how two random selections from a general 

 population will as a rule have organs of different means, of different variabilities, and 

 of different correlations, the means among themselves and the variabilities and corre- 

 lations among themselves forming systematic groups, it becomes obvious how any 

 assumption of the coefficient of correlation as a constant for local races runs wide of 

 the mark ; and this, whether natural or random evolution, is to be looked upon as 

 the source of the observed differences in character. What chiefly concerns the 

 biologist in this matter at present is this, that even a random selection of one organ 



