ORGANIC VARIABILITY 537 



is only of very limited value. From this distribution, or its 

 graph, we can obtain (i) the mean length ; (2) the modal 

 (or most prevalent length), which is not the same as the mean ; 

 and (3) the range of variation of length. But these constants 

 of the distribution are not given very approximately, and 

 some of them vary with the mode of grouping of the observa- 

 tions. The latter must be grouped and the choice of a group 

 is arbitrary. Different graphs thus arise, and in spite of the 

 honesty of the investigator he will usually exhibit (it may 

 be unconscious) bias in selecting the form of grouping. The 

 empirical distribution and graph must be replaced by the 

 theoretical ones . That is to say, we must find the limit to which 

 the observations tend were they to become indefinitely 

 numerous, or to embrace the entire population (which is being 

 investigated really by means of a sample). 



We take, as a first approximation to this result, the hypo- 

 thesis that the causes producing variation (or deviation from 

 the mean value of the independent variable), are a multiplicity 

 of small causes, all of which are independent of each other, 

 and the result of the operation of which is as likely to give 

 deviations in excess of the mean as it is to give deviations 

 in defect of the mean. Given these conditions it is now easy 

 (for the mathematician) to deduce the law of variability. Let 

 a pack of cards contain ten each of spades, clubs, diamonds, 

 and hearts, and let ten cards be drawn. 1 Let the cards be 

 reshuffled and drawn again, and let this be done (say) 200 times. 

 Then the probability of getting o, 1, 2 ... 8, 9, 10 spades 



is given by the successive terms of the expansion 200 (- + -) , 



for the chance of getting a spade, if one card be drawn is 



4 

 there being one spade to every four cards. The series given 



by trial is that first quoted on p. 535, and the series deduced 



by calculation is the second one. The general expression is 



M(p + q) n , where p is the probability of getting a certain 



result, and q is the probability of failing to get it : p + q — i, 



M = the number of trials, and n the number of different 



events (the numbers of spades that might be drawn in the 



example given). Now p and q may have any ratio between 



1 Note that to " draw " the cards so as to satisfy defined probabilities is not 

 so simple as it appears. 



