22 STATISTICAL METHODS. 



Curves of unlimited range on both sides: 



*))~fl 1)Q { 



Unsyinmetrical, y = y cos 0" e , where tan 6 = -, Type IV. 



CL 



2(7-2 



[Symmetrical, y = y e , the normal curve.] 



In these formulas : 



y = modal ordinate, to be especially reckoned for each type. 



y = the length of the ordinate (or area of rectangle) located at 



the distance x from y . 

 a = a part of the abscissa-axis XX T expressed in units of the 



classes. 

 e = the base of the Naperian system of logarithms, 2.71828. 



Curves of limited range are theoretically different from the 

 normal curve, which theoretically applies to cases where the classes 

 have an infinite range above and below the mean. Such an infinite 

 range is rare in biological statistics, although, as stated, the normal 

 curve often fits observational curves very closely. The range in 

 biological statistics may be limited at both extremes. Thus, the ratio 

 of carapace length to total length of the lobster is limited between 

 and 1. 



The range may be limited on one side only. Thus the ratio 



Antero-Post. Diam. . 



of a bivalve shell mar conceivably range from to 

 Dorso-Veut. Diam. 



co. The forms of the molluscan genera Pinna (or Malleus) and Solen 

 approach such extremes. 



Asymmetry or skewness is found in Type I (of which Type II is 

 the symmetrical limit), 'l^-pe III and Type IV. In skew curves the mode 

 and the mean are separated from each other by a certain distance, d. 

 Asymmetry is measured by a factor 



the result has the same sign as MS- 

 In Type I, A = ^ VPi~ -. 

 " " III, A = 14 V/F- 



To compare any observed frequency polygon of Type 

 I with, its corresponding theoretical curve. 



. x \?i/ , a:\i 2 



