V VII] Introduction \\ 



Integrating 



f H z x dx = N {TO' (A) - -i V&T3 (A) - 4= (& - 3) r 4 (A) 

 j - ( V6 V24 



or N$ (1 + a h ) = N {r f (A) - -4082,4829 Vft T3 (A) - '2041,2415 (fa - 3) T 4 (A) 



- -0912,8709 OS 3 '-10V&)T 6 (A)-...} (xi). 



On the other hand, if A be negative, a h changes sign and the even tetrachoric 

 functions change sign. Thus 



'-h 



z x dx = Nk(l-a h ) = N {TO" (A) - -4082,4829 



+ -2041,2415 (&-3)T 4 (A) - -0912,8709 (&' - lOVft) T 6 (A) + ...} ...(xi)6w. 



Here & = fi^/Hz 3 , $2 = pi/pf, where //, is the sth moment coefficient of the 

 distribution, and V& follows the sign of /* 3 . T O ' (A) is the i (1 + a), and T O " (A) is 

 the |(1 - a) of the probability integral of the normal curve. The T O (A) of our 

 table 





oo V27T J A V 2?r 



= (1 a) if A be negative, = 1 (1 + a) if A be positive, 

 as in the second integral. Accordingly : 



T / (A) = 1-T (A), TO" (A) = TO (A). 



We introduce T O ' (A) and T O " (A) to avoid tabling T O (A) throughout the range 

 from oo to + co . A like artifice is not required with other even order tetrachorics, 

 as they are all odd in A, and merely change their sign with A. This point must be 

 borne in mind when plotting z x from equations (x) and (x)bis; the second and 

 fourth terms, i.e. those in T 4 (A) and T 6 (A), will be negative for A negative. 



It is clear that equations (xi) and (xi) bis will provide the frequency of any 

 subrange of a distribution, by subtracting N$(l +a h ) from ^V|(l +a A'X ^ ^ a d 

 A' (A 7 > A) correspond to the limits of the subrange. For applications of these 

 equations, the reader is referred to pp. 289 290, 303 304 of Biometrika, 

 Vol. xvii. 



we assume that r g (h) and r 7 (h) are of the same order, that /3j is much larger than *22678 (/3 S ' - 



This is not confirmed by statistical experience. In other words the hypotheses from which Edgeworth 



started are invalid. See for example an illustration given in Biometrika, Vol. xvii. p. 227, with 



/9, = -4, j8 3 =36, #,' = 7-08350. 



In this case we must have -4r 7 (h) much larger than 17r 6 (ft), but as T B (h) can for some values of A be 

 4, 5 or even 6 times as large as T- (h), it is clear that there is no validity in neglecting the r t (h) term 

 in favour of Bdgeworth's term. 



