30 STATISTICAL METHODS. 



OTHER UNIMODAL FREQUENCY POLYGONS. 

 The formulas of Pearson's Types I to VI are as follows: 



Type I. ,= 



Type II. y= 



TypellL y=y l + j */<*. 



Type IV. y=y cos6 2m e-* e , where tan 0=j. 



Type V. y=y Q x- p e-r/ x . 



Type VI. y=y (x-l)^/x^. 



In these formulas: 

 x, abscissae; 

 2/ , the ordinate at the origin, to be especially reckoned for 



each type; 

 y, the height of the ordinate (or rectangle) located at the 



distance x from 2/ ; 

 I, a part of the abscissa-axis XX' expressed in units of the 



classes; 

 <?, the base of the Naperian system of logarithms, 2.71828. 



The other letters stand for relations that are explained in 

 the sections below treating of each type separately. 



The range of the curve is limited in both directions in 

 Types I and II, is limited in one direction only in Types III, 

 V, and VI, and is unlimited in both directions in Type IV 

 and the normal curve. The normal curve may give the best 

 fit, however, notwithstanding the fact that in biological 

 statistics the range is ordinarily limited at both extremes. 

 Thus the range of carapace length to total length of the 

 lobster is limited between and 1. The ratio of carapace 

 length to abdominal length in various crustaceans may, how- 

 ever, conceivably take any value from -f oo to 0. In the ratio 

 of dorsoventral to aiitero-posterior diameter the forms of the 

 molluscan genera Pinna or Malleus on the one hand and 

 Solen on the other approach such extremes. 



Asymmetry or Skewuess (a) is tound in Types I, III, 

 IV, V, and VI, In skew curves "the mode and the mean are 



