132 TEANS ACTIONS LIVERPOOL BIOLOGICAL SOCIETY. 



The interquartile range is calculated by finding the 

 " median " and " quartiles." The arithmetic is simple and 

 easy. The method is applied to the crude frequencies and it 

 is therefore affected by the errors of sampling and by the errors 

 that arise from grouping. If different methods of grouping 

 are adopted — in the cases of small distributions — different 

 medians and quartiles are obtained, and any one of them, found 

 by different methods of grouping, is equally probable. If we 

 group only in one way we get only one interquartile range, but, 

 obviously, there are other equally admissible interquartile 

 ranges which we have not calculated ! 



What measure of dispersion is to be adopted ? This 

 depends on the plan of the investigation for the measure in 

 question is only a means to some conclusion or other. Here 

 we adopt the measure called the " shortest half (or two- thirds, 

 or three-quarters) range." 



The Shortest H a 1 f - r a n g e . 



The total area of the frequency curve (or the sum of the 

 frequencies) is taken as 1,000 (for we are converting the 

 observed frequencies into " per-milles "). We take the sum- 

 mational curve figure and then take one-half of the total 

 vertical scale (or 500) on a pair of dividers and, with this, 

 measure off the distances aa l ,bb\cc l , etc., along the vertical 

 scale lines and from the points where the latter intersect the 

 summational curve. Thus we get the points a 1 , b l ,c l ,d l , e 1 , 

 and then (with a " french curve ") we draw a smooth curve 

 through them. 



Next, with a pair of dividers, we find the shortest distance, 

 measured along some horizontal scale line, between the curve 

 a\ &', c 1 , d l 9 e 1 and the summational one. Perhaps there are 

 several scale lines all sensibly the same in length and then we 

 approximate by finding the middle one. The places are 

 marked where this shortest horizontal line cuts both curves : 

 they are g and/ 1 in Fig. 6. From g and/ 1 perpendiculars are 



