\i. I'kari, and f. J. Dlinbar 



327 



Fol' thcse fi'eqiiency distributions tlic tollowing constants have boen determined : 

 iiK'uii, median, modc, Standard deviat.ioii, eoedficicnt of Variation, togcthor with tlic 

 probable crrors i)f thu tflii-eo most inipurtant constants, tho incan, thc Standard 

 deviation and tliu coelHcicnl uf variatrimi. Only an aii|iroxiniate vaiue of thc mode 

 was obtaincd, tbe nicthod uscd bcing tliat dcscribcd b}- Ynle*, wiicruin an a])proxi- 

 niation to thc vahic of thc modo is (k'tcrmincd from thc rehition : Modo = Mcan — 

 3 X (Meaii — Median)"!-. In detcrmining tlic vahic of thc Standard deviation from 

 tho equation, a = \/fi..,, Sheppard's correction was used in obtaining /a„. The formula 

 used in (ietcrmining thc ])robabic error of thc rocflHcient of Variation was that 



given by Macdonellj after Pearson, viz., p. E. of v = ± '0745 



i/2n 



1 + 2 



VIOO, 



00/ 



whcre v is tho cocfficicnt of Variation, and n is tiie nmnber of individuals. 



The vahics u( thcse constants for the frc(]ucncy distriljutions of Variation in tho 

 outer and inner diametors arc given in Table III. 



TABLE III. 

 Constants for Size Variations in 504 Individuais of Arcella vulgaris. 



Discussion of Results. 



Thoro are several points brought out by these constants i'egarding Variation in 

 Arcella, which seem to need some further discussion. 



Taking first the constants defining the type of tho species with rcspect to the 

 characters under consideration, it is seen that the mean and the modc are separatod 

 by an appreciable distance, or in other words the distribution is sivew. In the 

 case of the outer diameter this distance amouuts to 3-22G niikrons, or TGl abscissal 

 Units. In the caso of the inner diameter the distance bctwcen mean and mode 

 is "870 mikrons, or appro.xiniatcly '88 of thc absci.ssal unit for this di.stribution. 

 From thcse abs(.i!ute values of the diffcrcnces it wonld appcar that the distribution 

 for the outer diameter was markedly more skew than thc distrümtion for tho 

 inner diameter. It is to be remembered howcvcr that the means and class units 



* Yule, G. U., " Supplementary Note on the Deteinünatiou of the Mode." Jonr. Roy. Stat. Soc. 

 Vol. Lix. p. 398. 



t PItil. Trans. Vol. 186, A, p. 375. 

 X Biomctrika, Vol. i. p. 189. 



