1 Tables for Statisticians and Biometricians [V VII 



Now the forward difference formula is 



u e = w + 0&u - %6 (1 - 0) A 2 ^ + $0 (1 - 0) (2 - (9) A 3 M 



_ ^0(i - 0) (2 - 0) (3 - 0) A 4 w + etc. . . .(ix). 



But 0, as measured from 4'0, = '5275 ; 1 - = -4725. , 



.. Ti 9 (3-94725) = -000,5010 + '5275 (-000,3266) - -1246,21875 (- -000,0334) 

 + -0611,6857 (- -000,0856) - "0378,0982 (- -000,0101) 

 + -0262,5892 (- -000,0211) - -0195,7384 (+ -000,0053). 

 r 19 (3-94725) = -000,50100 - "000,00524 ^ 



+ -000,17228- -000,00055 6719. 



+ -000,00416 - -000,00010 

 + -000,00038 



The convergence is not very rapid, but the result is probably correct to a unit in 

 the last place. For most statistical purposes linear interpolation, i.e. 



Tl9 (3-94725) = -000,5010 + -000,1722 = -000,6732, 

 or a result correct to a unit in the sixth decimal place, would be adequate. 



Those who wish to describe frequency distributions not diverging too greatly 

 from the normal by tetrachoric series will find the fundamental formulae below 

 useful*. The first gives the ordinate z x of the frequency curve in terms of 

 the total frequency N, the standard deviation a x for the character x and the 

 tetrachoric functions of h = x/a- x . The second gives the area of the frequency 

 curve from h = oo up to a given value of h = X/<T X . 



If we denote this tail area by JV|(1 a h ), in accordance with the notation 

 of the probability integral of the normal curve, and write /3 3 ' for /8/V/8i, these 

 formulae are : 



(x), 



or * 9 m- (n (h) + -8164,9658 \/&r 4 (h) + "4564,3546 (& - 3) T 5 (h) 



+ -2236,0680 (&' - 10 V&) r e (h) + ...} ...... (x)fosf. 



* The method may be occasionally useful, but is not to be generally commended, because (i) the 

 series in practice does not invariably converge, (ii) impossible negative frequencies occur, and what is 

 of most importance, (iii) the resultant curve not infrequently exhibits sinuosities quite out of keeping 

 with our experience of the frequency distributions of homogeneous material. 



t The additional term introduced by Edgeworth as giving a closer approximation to the value of z x 

 than the terms to r 6 (h), namely, 



h r 7 (h) = + -9860,1330ft T, (h), 

 is, we believe, idle. It assumes that this term is markedly larger than -2236,0680 ($,' - 10\/ft) T 6 (h) or, if 



