124 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



Repeat this sampling again and the same misrepresentation 

 occurs, but it is different groups that are under- and over- 

 sampled. 



If this were all we could apply Pearson curves to such 

 data as are here given. But the theory supposes that the 

 population that is sampled is an homogeneous one in respect 

 of the characteristic that we measure. One could not legiti- 

 mately measure the stature of all the individuals in a crowded 

 church, say, and then base a Pearson curve on the results. 

 A number of the people are full-grown men, others are full- 

 grown women, and others again are boys and girls of different 

 ages. Thus there are groups in this church population, and the 

 mode of variability from the average is not the same in every 

 group. We ought to measure and classify the full-grown men 

 separately from the women, etc., making a separate curve for 

 each. The assemblage is an heterogeneous one. 



All fishery samples are, in general, heterogeneous, consisting 

 of fish of one, two, three, etc., years of age. We find this by 

 examination of a sample. It is, in general, quite impracticable 

 to attempt to separate the sample of plaice caught and measured 

 into its year-classes. Therefore we cannot (in general, again) 

 apply the method of Pearson curves to treatment of the 

 statistics, and this is fortunate, in one sense, because the 

 arithmetic that is involved is " colossal." Still there are 

 samples in which one year-class may preponderate so greatly 

 as to smother all the others. So some of the distributions in 

 this report have been " Pearsonified," with the object of 

 illustrating this discussion. 



The Construction of Summational Curves. 



The method that has been adopted has been to smooth 

 the observed frequency-distributions by making summational 

 series from them. Then all the information required is obtained 

 from the latter curves. The methods actually used will best 

 be described by an example. 



