176 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



method of integration enables us to " smooth " the data, for 

 a curve can be more easily drawn among the points repre- 

 senting the integrated frequencies than among the points 

 representing the actual frequencies themselves. The curve 

 can then be pricked with the point of a needle where each 

 ordinate crosses it and the corresponding point on the 

 vertical axis read off. These latter points are percentages 

 at and over each unit of length, and by subtracting each 

 from the one immediately above it the original frequency 

 series is reproduced in a " smoothed " form. Smoothing 

 the original frequencies by taking the means of groups 

 of three — the ordinary method — is faulty, since it 

 assumes that the mean ordinate is equidistant from the 

 two ordinates, and this is not the case when the curvature 

 changes rapidly : obviously Simpson's Rule ought to be 

 applied to find the position of the mean ordinate in the 

 application of this method. But in the integration of 

 the frequencies the irregularities become smoothed out 

 since plus and minus deviations are algebraically summed. 

 The integral curves are used to find measures of the 

 dispersions by graphic interpolation. The position of 

 the point of inflexion is first marked on the curve, and 

 vertical and horizontal lines are drawn to cut the corres- 

 ponding axes : the value of the mode is the number on 

 the horizontal axis cut by the vertical line. We may set 

 up as a measure of the dispersion the lengths on either side 

 of the mode between which and the mode 25 % of all the 

 fish in the sample lie; that is, we find the lower limit of 

 length by adding 25 to the percentage opposite to the 

 mode, and we find the upper limit of length by subtracting 

 25 from the modal percentage. Horizontal lines are then 

 drawn from these numbers on the vertical axis so as to 

 cut the curve, and then vertical lines are drawn from the 

 points on the curve thus obtained so as to cut the horizontal 



