lxii Tables for Statisticians and Biometricians [XXXV — XLVI 



Table XLII, (a) — (d) gives the values of /3 8 , /3 4 , /3 6 and /3„ in terms of /?, and 

 /3». Hence as soon as /3, and /3 2 are calculated we can find the numerical values of 



& = WsIh-2, &"*/%//*»*. 0, = ftuslfif, /3 = fi s /fi 2 4 (lv), 



theoretically. Although these values will not be those which would be absolutely 

 deduced from the data themselves, they will, considering the large probable errors 

 of /i 5 , fi e , ft? and fi s be reasonable approximations to them. The values of the 

 probable errors of /3, and /3 2 are determinable by formulae involving /3,, /3 2 ... /3 8 . 



From these formulae, Tables XXXVII and XXXVIII, giving the values of 

 V-ATXp, and ViVSp, have been constructed. Hence multiplying by ^ from Table V, 

 we obtain 



•67449 - , -67449^ 



-—2^ and — j=r S ft 



ViV Pl >JN 



the probable errors of /3, and y8 2 . 



If we add to the standard deviations of /8, and /3 2 , the correlation between 

 deviations in /3, and /3 2 , namely R^p,, which correlation is given in Table XXXIX, 

 we can find the probable errors of any functions of /3, and /3„. Two such important 

 functions are the distance d from mean to mode and the skewness sk of the 

 distribution. The probable errors of d and sk can be found from Tables XL and 

 XLI respectively, the former by multiplying the tabulated value viVSd/cr by crx^ 

 (from Table V), and the latter by multiplying the tabulated value \/NS S k by %i 

 (from Table V). 



Thus far we have only been concerned with the constants which describe 

 certain physical characters of the frequency distribution without regard to the 

 type of curve suited to the distribution. We now turn to the latter subject. 



It is known that the type of frequency depends upon a certain criterion k. 2 . 

 Hence near the critical values of « 2 more than one type of curve may describe the 

 frequency witliin the limit of the probable error of « 2 . Table XLIII gives the 

 probable error of * 2 , if the entries in that table be multiplied by the ■& of 

 Table V. 



The following are the series of Type curves which arise according to the value 

 of the criteria 



«,-2&-8A-6 (Ivi), 



, ft (ft + a? ( ivii) 



/9 2 is by necessity >f&. Hence for our curves all possible values of f3 u /3 2 lie in 

 the positive quadrant between the lines /3 2 = J/3, and & = -^-/3, + 1, the latter being 

 if we go to /3 8 the limit of failure of Type IV, for its fi 6 becomes infinite. Beyond 

 the latter line distributions are heterotypic. 



