XXIV 



Tables for Statisticians and Biometricians 



[IX 



Table IX (pp. 22—23) 



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

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



The nth incomplete normal moment function is defined to be 



fin («) 



\/2tt 



r x n e 



Jo 



■Jx2 



dx 



.(XV). 



We take 



m n {x) = ii n {x)j{{n— l)(ft — 3) (» - 5) ... lj if n be even) 

 = fi n (*)/{(« - 1) (h - 3) (« - 5) ... 2) if n be odd I 

 and m n (x) is the function tabled. 



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



-4x 2 \ 



z = 



{2Tr)i' l <r 1 (T.,...a- n \ r R 



e 



where 

 and 



X s = ^ {SiRgpXjl**) + 28' (RpgXpXg/apag)} j 



1, 



J *12 > **13 • • • ?'l 

 1 , ?'23 ... 7"o 



' 111 > ' M 



I'n-i > M 



..(xvii), 

 .(xviii), 



while jRpp and R pq are the usual minors. 



^ 2 = constant is the "ellipsoid" of equal frequency in n-dimensional space. 

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



I x =( X zdV 



Jo 



and 



r ,,, V27rit„_i (y) c i 



IJN = cT-rjr i o\ lf n be even 

 Xl 2. 4.6 ... (n — 2) 



W.(x) 



.(xix). 



if « be odd 



] .3. 5.. .(ft -2) 



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 I X JN = \, we obtain the 'ellipsoidal' contour y,o within which half the 

 frequency lies. This x<> 1% fcne " generalised probable error " of Pearson and Lee. 



