204 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



while discriminating between the sexes of Irish Sea plaice 

 up to Age-Group III. Thus the mean length of male fish 

 of Group II is 21' 03 cms., while that of females of the 

 same mean age is 21*26, a mean difference of 0'23 cms. 

 Now small as this difference is, it is significant, for the 

 number examined is large. Thus the standard deviations 

 (a/ fi 2 ) for the males and females of this group are + 3*11 

 and + 2 -87 respectively, and the corresponding " probable 



errors " of these means (0*6745 —=, a 2 being the " second 



moment " of the group frequency distributions, and N the 

 numbers of specimens) are + 0*052 and + 0*05 cm., so 

 that the probable error of the difference of the two means 

 ( Y (EJ 2 + (E 2 ) 2 , E x and E 2 being the probable errors) is only 



+ 0*022 cm., thus being very much less than the observed 

 difference. The difference of the mean lengths of the fish 

 in Group III is also very small — the figures are 29'69 for 

 the males and 26*58 for the females — and it is in the 

 opposite direction, the males being apparently slightly 

 longer than the females of the same mean length. But 

 we find, on calculating the probable error, that it is greater 

 than the observed difference in the means, so that the 

 number of plaice examined is not great enough to yield 

 satisfactory determinations of the mean lengths of the 

 sexes. This, obviously, is also the case with the fish of 

 Group IV. We may conclude, then, that the differences 

 between the mean lengths, or growth rates, of plaice up 

 to the end of the fourth complete year of age is small, 

 and need not be regarded as of practical importance. 



This being so, we can combine the sexes and thus 

 obtain as large series of figures as possible, in the hope 

 that satisfactory frequency distributions for each of the 

 age-groups may be obtained. In this way the series of 



