126 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



way : 1,000-2 °/ 00 fish are 13 or more than 13 cms. in length ; 

 999-7 % are 14 or more than 14 ; 993-3 % are 15 or more than 

 15 cms., and so on. Or, again : 263-2 °/ QO plaice are 22 cms. 

 or over 22 cms. in length and 506-8 °/ QO are 20 or over 20 cms. 

 long. Therefore 506-8 - 263-2 =- 243-6 (that is, about one- 

 quarter of the entire catch) are over 20, but less than 22 cms. 

 long. 



Col. (5) in the table, " y" represents the theoretical 

 frequencies as calculated by Pearson's method of curve-fitting. 

 Now let these theoretical frequencies be summed in the same 

 way as Col. (3) has been obtained : we thus get Col. (6), " 2^." 

 It will be seen that this is very similar to Col. (4), which gives 

 the result of the summation of the crude frequencies. The 

 crude 2/ 's are more like the theoretical 2/ 's than the crude 

 / 's are like the theoretical/ 's, and this is because, in the process 

 of .summation we have automatically got rid of the errors of 

 random sampling — or, at least, to some extent. How this 

 comes about is easily seen : if one class, say the fish of 18 to 

 19 cms., is over-represented in the sample, then all the other 

 classes will, on the average, be under-represented. Now in the 

 summing we add together at each stage over- and under- 

 sampled classes, and so the error of random sampling, apparent 

 in the frequency series, tends to disappear from the summa- 

 tional ones. Therefore, in graphing these various columns it 

 will be seen (fig. 3) that the crude and theoretical summational 

 series are nearly the same. 



From the summational series, made in this way, smoothed 

 frequency series can easily be constructed. First of all the 

 summational series must be graphed on a fairly big scale. The 

 curve must not be drawn free-hand, but by means of some 

 device that enables us to lay a spline, or steel spring, evenly 

 among all the points plotted. The curve should pass as nearly 

 as possible to all the points, but without necessarily passing 

 actually through any of them. It should be drawn by running 



