122 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



" Smoothing " might be effected by taking overlapping 

 averages. Thus, iustead of the frequency, 5, at mean length 



11*5, in the above example we might take — = 6»7 ; 



instead of 13, at mean length, 12*5, we might take 



~ = 21-3 and so on, all through the series. In 



some cases this method has an approved basis ; it means that 

 we are generally in doubt that any fish we measure is properly 

 measured : it may realty belong to the group in front, or that 

 behind the group in which we have placed it. This is so with 

 a number of fish in every sample. If one is very near 11 cms., 

 say, it may be really a little less than 11, so little less that 

 our necessarily hurried methods may not enable us to be 

 sure. But it is only a few of the fish about which we are in 

 doubt, in this way. That means that we ought to employ a 

 smoothing formula of this kind, 



fn-i + m fn+fn + l 



2 + m 

 where m is a small number, say 2 to 5. 



Pearson Probability Curves. 



The really scientific way to smooth such series as we have 

 is to calculate a theoretical distribution and then use this instead 

 of the crude series obtained by the measurements. Pearson 

 curves are based on the theories of probability. The different 

 results that are got in playing games of chance are explained 

 by assuming that these results are due to the operation of a 

 great number of small causes. The number of sixes one gets 

 on throwing a dozen dice at the same time, or the number of 

 heads we get when we throw a dozen pennies into the air, are 

 chance effects due to a great number of small, independent 

 causes, which are usually beyond our powers of control. The 

 theory of probabilty enables us to calculate such chance results 

 beforehand, and the calculated result agrees surprisingly with 



