88 
Higher Order Normal Product-Moments 
(j) s or t = 9. 
P9,2t+i= P2«fi,9 = 1-3.5 (2;+ l)r{945 + 2520ir2+ 1512^(/- 
+ 288i (^ - 1) (/ - 2) r« + 16i (i - 1) (/ - 2) - 3) j-s}, 
Ps. 9 = 945r (945 + lOOSOr^ + 18144r« + 6912r6 + 384r«). 
{I) SOTt= 10. 
Iho.^t = Iht. 10 = 1 • 3 . 5 (2i - 1) (945 + dimtr^ + 12600i (/ - 1 ) r* 
+ 5040< (/ - 1) - 2) + 720< - 1) - 2) - 3) 
+ 32<(i 2) (i-3) (^-4) 
TJio.io = 945 (945 + 47250/^ + 252000r4 ^- 302400r« + 864000r8 + 3840ri«). 
The table on pp. 90-1 gives the numerical values of these coefficients. We 
proceed to illustrate their use. 
Illustration I. In discussing the relation of auricular height (y) with age (x) 
of a girl's head a sample of 2272 individuals was found to provide the following 
product-moment coefficients : 
= 3-113,712, 93,1 = 74-447,616, 
q2,i = - 1-957,022, 94,i = - 108-701,559. 
Are these incompatible with normal correlation? (See K. Pearson, On the General 
Theory of Skew Correlation and non-linear Regression, Drapers' Company Research 
Memoirs, Biometric Series II, p. 35.) We have 
= 3-064,819, a, = 3-454,125, 
and r = -294,128, 
and the leading subscript above corresponds to the x coordinate. We need first 
the values of g'g.i' Isa ^^^^ Via hypothesis of normality. Clearly §'2,1 ^i^d q^-i 
will be zero, and using linear interpolation : 
93,1 ~ (^x^^yP 3,1 
= 99-437,979 x -88256 
= 87-759,983 = 87-7600, say. 
In the next place we require the probable errors of these ^''s. The general 
expression for the probable error of a product-moment about the mean is given 
in Biomelrila, Vol. ix, p. 3. In our present notation it is 
Now remembering that for a normal distribution q vanishes when s -|- ? is odd, 
and that ^'4 0 = 3ct/, while s + t 
?s,i= 10 ^ v's,t<yx'(^v, 
we have 
^^V. = '74,2 + ^ga.o'Z'l,! + + 4g\i?2,0 - - 2?2,2'72.0 
= a/a/ {lOp'4,2 -f 8r2 -M - 4rp'3,i - 
a,. , = ^ {10?/4,2 + + 1 - 4r?/3,i - '2f\J (x). 
