246 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



to 50-24, 50-25 to 50-49, 50-50 to 50-74, 50-75 to 50-99, etc. 

 The range of the classes is now doubled, and so the risk of 

 mis-measurement is minimised. 



There may also be errors due to distortion (as the results 

 of bad condition, icing, freezing, etc.), but we do not consider 

 that this source of error is considerable. 



The error of random sampling is the greatest, and it cannot 

 be avoided except by making very large series of measurements 

 and on big samples taken from the same locality at the same 

 time. This is, in general, impracticable. The method used is 

 based, therefore, on the study of small samples (50 or there- 

 abouts). These are subject to errors of measurement, 

 distortion and sampling. All this has to be considered in making 

 conclusions as to the existence of " races." 



In the following pages use has been made of Pearson's 

 method of " goodness of fit " (see Biometrika, Vol. VIII, 

 pp. 250-254). We make the hypothesis : 



All Manx fish belong to the same race and the. differences 

 between sample and sample are due to errors of random sampling. 



This hypothesis is now to be tested. 



Let there be two samples taken at the same place, but at 

 times a month apart. For instance (Table XII, D) : — 

 D = 51-00—51-25, 51-25—51-50, 51-50—51-75, etc. 

 / = 23 24 36 etc. (June). 



f 2 = 14 , 12 , 21 , etc. (July). 



Consider only the frequency, 24, corresponding to the class- 

 range, 51-25—51-50. 



Suppose that, instead of only one sample for June, we had 

 had five samples : then, instead of the one frequency, 24, we 

 might have had (say) 19, 23, 24, 26, 28, or some such numbers — 

 in general, we should have had about as many different 

 frequencies as there were samples. This is because the range 

 of D = 51-25 — 51-50 is only one part of the whole range, 

 which is about 46 to 59, and our methods of collecting are such 



