134 TRANSACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



dropped to cut the horizontal axis, and these latter points, 

 read off on the scale of lengths, give the abscissae of the shortest 

 half -range : the latter is about 18-5 to 21-5 cms., and within 

 this range of lengths one-half of all the fish in the catch are 

 contained. 



We now go further. From the point $r", on the vertical 

 scale, 150 units are measured downwards getting the point h u . 

 A horizontal line h u h is drawn to cut the summational curve 

 in h and a perpendicular M 1 is drawn to cut the horizontal 

 axis in h l . Between g { and h\ that is, between about 21-5 and 

 23 cms., another 15 per cent, of all the catch is contained. 

 Next we prolong the vertical line,/ 11 / 1 , upwards to cut the 

 summational curve in/, and then a horizontal j/ 1 " is drawn 

 across to cut the vertical axis in/" 1 . From the latter point 

 150 units are measured upwards on the vertical axis giving a 

 point on the latter at about 1,000. Therefore a further 15 per 

 cent, of the fish are contained within the range of lengths 

 14 and 19 cms. 



Summarising we find : 



One-half of all the fish are over 19 and less than 



22 cms. in length ; 

 80 per cent, of all are greater than 14 and less than 

 23-5 cms. in length. 



Obviously we can extend the method. We might repeat 

 the above construction, using two-thirds of the vertical scale 

 length, or three-fourths : these figures would give us the 

 shortest two-third and three-fourths' ranges. Or we may take 

 the point 20 cms. on the horizontal scale, draw a perpendicular 

 upwards to cut the curve and then a horizontal across to cut 

 the vertical axis in the point 600. That shows that 60 per 

 cent, of all the fish are 20 cms., or more than 20 cms. in length. 

 Or, again, we may find the mode and then draw a horizontal 

 across to cut the vertical axis. Stepping off 25 per cent, of 

 the latter scale on either side we get two additional points. 



