xxx Tables for Statisticians and Biometricians [XI 



Now on the assumption of a normal distribution : 



Jx v2tt<t 



m JV<7 S | <**«" ***•<&/ 



J x' 



= iW {/x s (so )-/x s (*')}• 

 Here Table IX (p. 22) shows that if s be odd, m, (oo ) = '398,9423, i.e. 1/V2T, 

 and (p. 23) if s be even, m, (00 ) = -500,0000. 



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

 whole population, m s (x') should be subtracted from '398,9423 or from '500,0000 

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

 s is odd or even respectively. It is convenient to term fi s ( 00 ) — ft, (x') the com- 

 plementary incomplete moment function of order s*. 



For s = 1 and s = 2, we have 



W/ij' = criV {?»! (00 ) — mj («')}, 



«/f/ = o- 2 iV [m 2 (00 ) — ma (#')], 



for in this case the multiplying factors to proceed from m s {x') to /x,(* - ') are both 

 unity. 



Now cc = x/a 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 : 



"IT, 



d = /*/ — A'o" = 



a 



{OTj (00 ) — Ml] (x)) — ti 



.(xxvi). 



N 

 2* = <r> — {m 2 (00 ) - m, (a;')} - ft? 



Of course m, (oo ) — m l (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] - \ (-5905) (-4095) [26358] 



= -039,9222, 



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



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



Similarly from Table IX (p. 23) : 



m, (-45905) = '008,1136 + 5905 [73,162] - 1 ('5905) ('4095) [30661] 



= 012,0630, 



and m 2 (00 ) - m, ('45905) = '487,9370. 



* It is the function used by Dr Alice Lee and myself, Biometrika, Vol. vi. p. 65 to form Table XI, 

 p. 25, but by an oversight not adequately distinguished in symbol from /», (x') of p. GO of the same memoir. 



