180 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



the theory of the normal curve. The latter has a precise theo- 

 retical significance; that is, the deviations from the mean 

 value are those very numerous, small, uncoordinated ones, 

 as likely to cause positive as negative deviations, and the 

 total effect of which is as likely to act in one direction as 

 the other. This is not the case with the distributions which 

 we have to consider in fishery statistics. The material 

 sampled is not homogeneous, for we may have to deal with 

 " races," or groups of fish which have different growth- 

 rates, for instance; the samples taken do not accurately 

 represent the natural population, and would not, however 

 numerous they were, for the apparatus of collection — the 

 trawl-net — selects from the population ; and even if the 

 sample taken were a truly representative one, it would not 

 give us a symmetrical frequency distribution. The fact is 

 that in dealing with length-frequency fishery statistics, we 

 are not dealing with distributions expressible by the 

 normal curve of error, nor even by any of the Pearson 

 probability curves. 



Such being the case, it seems that such conceptions as 

 standard deviations, means, medians, and quartiles are quite 

 artificial ones, and their use does not enable us to analyse 

 the material studied with success. What we have really 

 to consider is the real, " natural " form of the distribu- 

 tions : it is the mode and the deviations from the mode 

 that we have to express by some figures, and not the mean 

 or median and the deviations from these positions. The 

 distribution graphed in Figs. 1 and 2, (Nelson Buoy, 

 August, 1909-1913), is only a moderately asymmetrical 

 one, other distributions may be much less symmetrical. 

 I have shown the positions of mean, median and empirical 

 mode in this diagram, and it will be seen that if we calcu- 

 late the quartiles from the median, or the standard 

 deviation, and the probable error from the mean, we obtain 



