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



characters be not always in the same direction. Systematic change of characters 

 produced by random selection may be spoken of as random evolution. Random 

 evolution is -theoretically a possible cause of systematic change ; experiment only can 

 determine how great is its effectiveness in differentiating local races. 



(77) In the case of a normal distribution of variation defined by the mean h and 

 the standard deviation cr, it has been usual to suppose that the error made in the 

 mean is independent of the error made in the variation. In other words, it has 

 been assumed that G,,, vanishes, although no proof has been given, or possibly it 

 has not been realised that a proof was necessary. In this case f is of the form 



r 2/0 21 j i 



expt. [ar/2crj and - - == , whence 



V/(27r) a- da dx CT" ' 



G = 2 (*"4 expt. - {^ 2 /2o- 2 l - - dx = 0. 

 J-oo a ' v / ( 2 7r)o- 



Thus there is no correlation between error in the mean and error in the standard 

 deviation. This assumes that we stop at the square terms in (vi.). If, however, we 

 include the cubic terms, &c., product terms in A/i and ACT do arise, and we cannot 

 state straight off that no correlation exists, although it may be very small. In the 

 case of all skew variation, such as is so frequent among plants and animals, a corre- 

 lation will always be found between deviation or error in the mean and the like in 

 the standard deviation. In other words, to alter the mean by selection (artificial or 

 random) is to alter the variation of an organ. 



With the exception of the statements in this paragraph (17), the whole of our 

 general conclusions in this section are independent of any particular law of frequency. 



(4) On the Determination of the Probable Errors and the Error Correlations of the 



Frequency Constants. 



Let iji, 7)2, f] 3 , . . ., be the frequency constants, whether they be the means, standard 

 deviations, or correlations of a complex of organs. Then if we neglect cubic and higher 

 terms in the deviations A^ 1} A^, A^ 3 , . . ., the frequency surface giving the distribu- 

 tion of the variations in the deviations is 



P A = Po expt. - l [S {,,(A>?,) 2 } - 28 K A^A^}], 

 where 



It is required to find 2,., the standard deviation of AT;,., and R, s , the coefficient of 

 correlation between A??, and A^. 



