544 SCIENCE PROGRESS 



clearly to be regarded as the resultant of a multiplicity of 

 contributory causes, and its treatment thus becomes a matter 

 for the application of statistical principles. Conceding this 

 we have now to find what general results emerge. Let us 

 note, first of all, the difference between the typical biological 

 frequency distribution and the physical curve of error. The 

 latter arises when a series of determinations of some physical 

 constant are made in such a way that constant errors of ex- 

 periment and errors resulting in bias are avoided. Even then 

 divergent values for the constant must be obtained as the 

 result of inevitable experimental errors, but the methods of 

 determination are so -designed as to make it equally probable 

 that the error will be in excess of the mean or in defect of the 

 mean. There is an unique, natural value to be obtained and 

 the deviations from this observed in individual experiments 

 have no reality (in a sense). The mean is the most probable 

 value of the constant, and the deviations are a measure of 

 the experimental error. But in the biological frequency dis- 

 tribution the mean is purely an abstraction. There is a 

 modal, or most prevalent frequency, or value of the character 

 studied, and this modal value is, in general, not the same as 

 the value of the mean. Further, the deviates from the mode, 

 or from the mean, are all real natural things. If we compare 

 a biological frequency distribution with a physical curve of 

 error we must think of the individuals of an elementary species 

 as the results of an experiment. There ought to be unique values 

 for each character, but imperfections in the developmental 

 methods have given rise to the variability. The idea is highly 

 artificial, and without considering where it may lead we can 

 hardly accept it. 



We do not, in the least, know what are the contributory 

 causes that produce variability, but we can find what is their 

 resultant in any particular case from the mathematical analysis 

 based on Pearson's methods. The generalised probability 

 equation (5) gives us, as we have seen, three cases : (1) when 

 the coefficients c x and c 2 vanish, when we obtain the " normal 

 curve of error." The contributory causes of variation are now 

 numerous, they are all independent of each other, and their 

 result is as likely to produce variations in excess of the mean 

 as it is to produce variations in defect of the mean. That is 

 to say, the variations are " fortuitous " and without tendency, 



