XXIV 



Tables for Statisticians and Bionietricians 



[IX 



Table IX (pp. 22—23) 



Values of the Incomplete Normal Moment Functions. (Calculated by Dr Alice 

 Lee, Biometrika, Vol. vi. p. 59.) 



The /itli incomplete normal moment function is defined to be 



If'' - ' t- 

 /^» (*■) = 7ii= *"e '-• dx (xv). 



V air Jo 



We take 



/H„ (*■) = /i,i {x)l[{n — 1 ) ()( — o) (h - 5) . . . 1 ) if n be even 



= ya„ {x)l{{n - 1) (n - 3) (h - 5) ... 2j if n be odd 

 aud nin (.r) is the function tabled. 



In multiple correlation (supposed normal), the frequency surface is 



.(xvi), 



N 



(27r)i"o-i(r, ...o-„VS 



-hx^ 



where 

 and 



%"- = ^[^ (Rpp'^p'/o-p') + 2*8' {Rpq00pXg/<7p<T,j)} 



R = 



1, ri2, »-i3...r-i 

 ?•„,, 1, Vos ... r„ 



'Vn ••■ 1 



..(xvii), 

 .(xviii), 



while Rpp and Rpq are the usual minors. 



T^- = constant is the "ellipsoid" of equal frequency in n-dimensional space. 

 The total frequency, i.e. the volume of the surface, inside any ellipsoid x is 



Jo 



2CIV 



and 



/,/i\^=.^^M^. ifnbeeveni 



2.4.6...(«-2) 



2yan-i(x) 

 1 .3.5...(»-2) 



.(xix). 



if n be odd 



Thus a knowledge of the incomplete normal moment functions enables us 

 to predict for multiple variables whether an outlying observation consisting of 

 a system of n variate values is or is not reasonably probable. 



If Ij^JN = \, we obtain the 'ellipsoidal' contour ■)(_„ within which half the 

 frequency lies. This Xo i** ^^^^ "generalised probable error" of Pearson and Lee. 



