Karl Pearson and David Heron 
303 
and let this distribution be the real population or an absolutely proportional sample. 
Now for this population with the scheme < ~r^, we nave a d ~ be = °> an< l a ^ our 
measures of association vanish. If we take a sample of ri from such a population 
of n, the value 77 = (ad — bc)/n 2 will not for the sample be zero, but, if 
rj' = (a'd' — b'c')jn 2 , 
where a', d' , b' and c' are the values in the sample, will have a standard 
deviation* 
(a + b) (a + c) (d + b)(d + c) 
X \In' ' 
Now let us compare this with 
(a'+b')(a +c')(d' + b')(d' + c') 1 
»' 4 \lri ' 
the value of oy derived from the sample itself. Now to units the value of the 
sample might be 
TABLE II. 
A 
Not-.l 
Totals 
B 
971 
23 
994 
Not-5 ... 
6 
0 
6 
Totals 
977 
23 
1000 
and it is clear that and oy will be exactly the same, as they depend only on 
the marginal totals. Thus 
1 
V-994 x -977 x -006 x -028 
V1000 
= 000366. 
But j] = "000138. Accordingly (ad — bc)jN- when d = 0 is not significant, having 
regard to its probable error, and the association is zero. On Mr Yule's theory 
Q = — 1 and its probable error is zero. 
Now the standard deviation of Q, for Q zero, is 
1 (a -+- c) (c + d) (6 + d) (a + b ) 
(ad + bef 
for a sample of n', where a, b, c, d refer to the original population. 
Let us consider the differences for the above material which will arise from 
calculating 0 o-q on the population in Table I and on the sample in Table II. On 
Table I we find O o- Q = T37 ; on Table II 0 cr Q / = 2-62. Hence either Table is 
* Pearson, "On a novel method of Kegarding the Association of two Vaiiales," Drapers' Research 
Memoirs, Biometric Series, No. vm, Dulau & Co., 1912, p. 7. 
