Karl Pearson and Egon S. Pearson 
135 
between the two. A linear interpolation will probably suffice in most cases to 
determine r with sufficient accuracy. 
It will be observed that what we are trying to do is to fit a normal correlation 
surface to a series of cell frequencies. We may do this by equating product- 
moments, or actual cell frequencies properly weighted. The factors ~ and ( ;r 
come into our equations as a form of weights. When is small as compared with 
that cell will contribute less to the general equations for r, and when «ss'' is 
large as compared with Hjs', the contribution will be considerable. If the observed 
results were closely normal then n^^' would be nearly w^s'. If we might assume the 
differences of '/is^.- and ri^y so small as to be negligible we should have : 
r=S {%T„^,T: + rXT,X.r:+ ...+rP'^,Tp%.T;+ ... (xvi) ter, 
and 
0 = *S' C^sTi^s't/ + 2rS)-,ToSr,'T; + ... + jj)*-i^sTp^,'Tp' + . . . (xvii) bis, 
instead of (xvi) bis and (xvii). These equations it will be found are identically 
satisfied. Hence our values for r from (xvi) and (xvii) depend on rigg' differing 
from flss'. 
(3) We now proceed to illustrate the application of these results. 
Stature of Father and Son. 
The following table gives a correlation table for the inheritance of stature in 
Father and Son made up in broad categories corresponding to eye-colour groups*. 
Upon this material we shall be able to test our correlations and our graph against 
those found by definite numerical groupings. 
Stature of Father (Broad Categories). 
T3 
a 
p 
' a: 
02 O 
=4-1 ^ 
/ 
a 
3 
4 
6' 
7 
Totals 
]' 
4 
22 
7 
1 
34 
23 
154 
84 
26 
8 
6 
301 
o 
8 
87 
75 
66 
22 
24 
2 
284 
V 
1 
29 
36 
37 
14 
14 
6 
137 
5' 
18 
27 
26 
11 
18 
5 
105 
ij 
9 
26 
19 
7 
29 
8 
98 
"i 
3 
9 
6 
6 
10 
7 
41 
Totals 
36 
322 
264 
ISO 
69 
101 
28 
1000 
The positive direction of x is Iroiu left to right and of y vertically downwards. 
It will siiffice to take the t's to five decimal figures but it will be needful to go 
further with the t's if the T's are to be taken correctly to five figures from (viii). 
The general reduction formula for the T's is : 
Ti_i {x) {t - 2) Vrn. of' - Ti_, (x) (t - 3) {{t -l)x'' + l) 
or, 
9« 
V«(i-l)(«n^-2) + l) 
T,_, {X) - {or (i - 1) + 1) ^-^2 T,_, {X) 
* See Biometrika, Vol. ix. p. 220 
. .(xviii), 
(xviii) bis. 
