288 PROFESSOR K. PEARSON AND MR. L. N. G. FILON 



- r/ '-'' + #- S "" R """ ~ ffi2"F'>j} '''" (cxiv.). 

 J^ -f SV/X.SA,, 



(cxv.). 



These results show us (hat it is, in the general case of skew variation, impossible to 

 select any one of the quantities mean size, range, variability, or skewness of an on/an, 

 inthout at the same time in all probability modifying all the others. 



For example, the frequency of the incidence of certain types of diseases at different 

 ages follows a distribution of this character. Hence, if any special class of the 

 community had a mean age of incidence differing from that of the general population, 

 we should expect correlated changes in such other characteristics of the disease as 

 (i.) its first appearance ; (ii.) its last appearance ; (iii.) its tendence to heavier 

 incidence above or below the mean age of incidence ; (iv.) the concentration of its 

 incidence about the mean age of incidence for this selected class. 



Precisely similar series of changes would arise in the case of a random selection of 

 individuals having the variability of a certain organ greater or less than that of the 

 general population, there would be correlated changes in the size, range, and 

 skewness of the distribution of this organ. 



Turning to the mean and modal frequencies, we have 



Ay, _ Ai , [ 3 _ _1 i I _ , dti(m, + m. t + 2) __ rS(? 1 + 1 



~ 2 w, + 7?J + 2 " 2 ?, + 1 " ' r?Oi + '"'2 + 2 ) " " <l( m i + 1) riim ' 



_ JL <'S('i + * s + 2) 



" " 



1 a 7/f, + m., + 2 2 w, + 1 (f (?, + m, + 2) d (m, + 1 



/i,A?, -J- //._> Aw.,, say (cxvi.). 



Similarly, 



A?/., At f 1 1 1 rfl3(m, + 7/O rfS(m,)l 



-' = ; r i~ 7 + ;^ ~; ;, ~ + ' r " r A?n, 



i/j o [?! + ? +1 2 (m^ + wi) 2m l d(m l + m*) dm l \ 



' |m, +m, + 1 "^ 2 (m. + w s ) ~ 2v?7 2 "" ' ~d(m^~+ in,) ' ~ dm. J AWl '- 

 = A6/6 + ki Am, H- h Amo, say (cxvii.). 



Here /i,, /i 2 , 7c,, k. 2 can be easily calculated, if we note that 



. . (cxviii.). 



= T/p, where T is the same as on p. 272. 



