ORGANIC VARIABILITY 539 



curve is that long known in biology as the " normal curve of 

 error," and for a time it was very generally applied to de- 

 scribe the frequency distributions formed when organic vari- 

 ability was studied. 



It is very extraordinary that the " normal curve " should 

 have been so extensively used in biology. It must have been 

 seen that it usually did not fit frequency observations well, 

 while there was no justification for the assumption on which 

 it was based being generally applicable to variability in 

 organisms. As we have seen, that assumption is that the causes 

 producing deviations from the mean position of variability 

 are independent, and without tendency to either the positive 

 or negative sides of the mean. It must, over and over again, 

 have been seen that frequency distributions were, as a general 

 rule, asymmetrical — that is, observation of a great number of 

 empirical graphs shows clearly that the curve generally rises 

 steeply from zero to a maximum, and then falls less steeply 

 or vice versa, so that the mean and maximal ordinates are not 

 the same. Obviously (so it seems now) biologists ought to 

 have applied the continuous curve obtained by Karl Pearson, 

 as the limit to the skew binomials, in the same manner as the 

 Gaussian curve was obtained as the limit to the symmetrical 



1 +-)e- yX , and it represents 



the probabilities giving rise to a chance distribution when the 

 contributory causes of deviation from the mean are very 

 numerous and independent, and when they tend to give devia- 

 tions in excess of the mean, and in defect of the mean, which 

 are no longer equal to each other. 



But again something more general was required, that is, 

 an expression for the probabilities that arise when the causes 

 of variation may be either independent or correlated, and 

 may or may not give deviations in excess of the mean equal 

 to those in defect of the mean. An expression for such a 

 series may be deduced (by a mathematician) from " ele- 

 mentary " theorems in probability. It is 



pn(pn — 1 ) . . . (pn — r + O f , rqn 



n(n — 1) . . . (n — r+i) \ pn — r -f 1 



r(r — 1 ) t qn{qn — 1 ) 



[2_ {pn — r + i)(pn — r + 2) 



■■) 



