92 
Higher Order Normal Product-Moments 
The ^'s of the marginal distributions show a markedly skew and non-normal 
system. The regression is, however, closely linear. Discuss the values of the 
product-moment coefficients : 
(/2,i = 11-919,404, 
^1,2 = 15-598,613, 
92,2 = 401-523,496. 
For a normal system with the above correlation coefficient we should have: 
?2,i = '/i,2 = 0 and ^2,2 = <^/<^/p'2,2 = 362-192,761. 
Thus A^a,! = 11-919,404, 
A9i,2 = 15-598,613, 
A(72,2 = 39-330,735. 
We require to consider the probable errors of the g's, which are given bv 
•6744898 times the following standard deviations: 
. = ^ + + 1 - 4rp'3,i - 2?,'2,2}% 
^.„. - ^'{]02/2,4+ 8r2+ 1 - 4rpV3- 2p'2,2F, 
0,.. = ~^{\^^VUA-V'\i (xiii). 
We determine for the above value of r: 
^)'2. 2 = 2-217,5021, ?9'3, 1 = 3 = 2-340,6750, 
P'2,4 = ?''4,2 = 1-030,50246, ^"'4,4 = -617,239437. 
Our results for a normal distribution are : 
P.E. of r = -004,882, 
P.E. of<72,i= 1-091,473, 
P.E. ofgi|2 = 1-320,585, 
P.E. of 92^2 = 15-360,681. 
Hence A92, i/P.E. of ^2, 1 = 10-920, 
A9i,2/P.E. of f?i, 2 = 11-812, 
A92,2/P.E. of 92, 2 = 2-560. 
The deviations in the higher moment coefficients are at once seen to be markedly 
significant. But it will be noted that 92,2 in the previous case does not differ so 
markedly in value from the normal as the odd moment coefficients. It seems there- 
fore likely, when a distribution is markedly skew, but the regression linear, that 
the even-even product-moment coefficients will not differ widely from the normal 
values, but that the even-odd ones will do so. It is possible that this is related to 
the fact that in distributions (such as 3 x 3 tables) which can be reduced in various 
ways to a tetrachoric table, correlation calculated from regression line diagonal 
cells is usually far more accurate than correlation calculated from non-regression 
line diagonal cells. 
Equations (x) to (xiii) are of value beyond the present illustrations. Further 
uses of the above formulae and tables are provided in a memoir on "Generalised 
TchebychefE Theorems" which will shortly be published. We have to thank 
Dr Kirstine Smith for much help in the preparation of this paper. 
