MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 277 



2, = <r/v/(2n), 



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= = 



The first, second, fourth and fifth of these results are old ; the rost appear to bo 

 novel and of some importance. 



In the first place we notice that given a population which is really normal, we 

 should not expect a random selection to exhibit all the signs of normality. Its 



skevvness will differ from zero with a probable error of '67449 A/ (;j- ). For example, 



in a random selection of 600 from a normal population, the skewness will be as likely 

 to exceed as to fall short of '034. Hence an exhibition of skewness of less than once 



VI '* \ 

 ( ) must not in itself merely be taken to indicate an absence of 

 \2w/ 



normality in a general population. 



Again, in a random selection from a general population, the mode will differ 

 from the mean, even if the population be normal, with a probable error of 



G7449 A/ (-} o: Thus, in a population of 600, a difference between the mean and 



the mode of '0340- should not be taken to indicate want of normality. Generally, 

 the divergence between mean and mode in a population must at least exceed once to 



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twice '67449 A/i^) "* f r us to be able to argue on this ground alone that the 



population has not a normal distribution. 



Again, the third moment not being zero, but having a value of once or twice '67449 



X 2 A/ ( j cr 3 , is not in itself an argument for skew frequency. 



The above statements are an important addition to the second memoir of this 

 series ; they give us the criterion, there wanting, to distinguish between a skewness 

 which is characteristic of a population and one which might arise by the random 

 selection of a population of the given size out of a larger, but really normal, popula- 

 tion. 



(e.) We may now note the exceedingly interesting conclusions which these results 

 have for the theory of evolution. 



Suppose an organ to have, as so many do, skew variation, then we notice 



