370 Prof. Karl Pearson on the Influence of 



Thus far we have seen that a sensible modification in the 

 expected mean may be needful and most probably a still 

 more appreciable deviation in the variability about that 

 mean. It is necessary, however, to consider how far the 

 distribution is really normal round the mean. Otherwise 

 we have no justification for calculating the odds on the basis 

 of tables of the probability integral. 



The full values of the second, third and fourth moments 

 about the mean for the hypergeometrical series are given 

 in the paper already referred to (pp. 239-240), and with 

 the changes requisite to suit our formula (iv.) are : 



iL _m( p±l)(q + iy / _m-l\ .... (x. 



m(y> + l)( g + l) (y-ri/ 1+ !»^\( 1 + 2(^-l)\ (xi . 



^- (n + 2)* \ l+ n + SJV^ » + 4 f 



~l^+2p~ V n + 3J\ n + A m-2 n + 5)J 



+ (n + 2Y V + fi + 3/ V % + 4 \ »+5/7 



In any actual case there is no difficulty in calculating /^ 2 , 

 fjL 3 and //, 4 ; then, if /3 1 =^ 3 2 /^2 3 and ft = /^ 2 2 we can find 

 A and ft, and these ought to be sensibly zero and 3 respec- 

 tively, if the system is to have the symmetry and meso- 

 kurtosis, which characterize the Gaussian curve. 



The values of ft and ft can, however, be found directly 

 from the following formulae : 



p- (4-PT V + »+*J (xiii) 



Pl m(^ + l)(^+l) ■ , «-l 

 + ra + 3 



„,_1 / 8 _ _9\ 



_ ?jm(m 

 ^ = (» + 2) 



1 + 



» + 3 



m — 1/. ■»! — 2' 



(» + 2V » + -x, 



M+^ 



m(p + l)(q+l) 1 w-1 



i+ rc + 3 



