XXX Tables for Statisticians and Bionietricians [XI 



Now on the assumption of a normal distribution : 



= N<t' \ a-'e -•" " dx 



J x' 

 = iVo-* I/is (oo)-/^s («')). 



Here Table IX (p. 22) shows that if s be odd, in, (^ ) = -398,9423, i.e. 1/V27, 

 and (p. 23) if s be even, wi^ (« ) = 500,0000. 



Hence iu obtaining the moment coefficients of the tail, about the mean of the 

 whole population, ??is («') should be subtracted from ■398,9423 or from -.500,0000 

 before the results are multiplied by (s — 1) (s — 3) ... 2 or (s — l)(s — 3) ... 1, when 

 s is odd or even respectively. It is convenient to term fis{co)— /is{a;') the com- 

 plementary incomplete moment function of order s*. 



For s = 1 and s = 2, we have 



n/j,i = (tN [ini (co ) — vii («')}, 



ii/j,.,' = cr"N {m^ (oo ) — VI2 («')}> 

 for in this case the multiplying factors to proceed from ins{x') to /tis(.«') are both 

 unity. 



Now x' = xj(T can be found when n is known from Tables II or III. Hence we 

 have for the distance of centroid of tail from its stump, and for the square of its 

 standard-deviation about its centroid: 



d = fj,,' — ll' a = a 



{hIi(x )-»!,(«')} -W 



N 

 2^ = 0-- — {'/;;., (00 ) — 7», (W — a,'- 



I 



.(xxvi). 



Of course «i, (20 ) — m-^ {x) is the z of Sheppard's Tables II and III. 



Returning to our numerical example, we have from Table IX (p. 22): 



m, (-45905) = -030,6721 + -5905 [162049] -i (-5905) (-4095) [26358] 



= -039,9222, 



mi (00 ) - m, (-45905) = -359,02. 



Found directly from Sheppard's Tables, it equals '35905. 



Similarly from Table IX (p. 23) : 



nio (-4.5905) = 008,1136 + -5905 [73,162] -i (-5905) (-4095) [30661] 



= -012,0630, 



and m., (00 ) - /?i2 (-45905) = •487,9370. 



* It is the function used by Dr Alice Lee and myself, BiometriUa, Vol. vi. p. G5 to form Table XI, 

 1>. '25, but by an oversight not adequately distinguished in symbol from /u, (x') of p. 00 of the same memoir. 



