Karl Pearson and David Heron 
169 
other very low relationship between the attributes under investigation, is that 
" these small figures represent perfectly appreciable intensities of the product sum 
correlation r." But this is an entirely different matter*, the intensity of correla- 
tion on the scale 0 to 1 is quite distinct from its degree of reliability. Mr Yule 
would have termed '95 a very high association had it been deduced from 1000 cases, 
and we term '01 a very low correlation even when deduced from a census population. 
Here are the results for which we have calculated the probable errors - ]". 
* Mr Yule himself says (loc. cit. p. 651) that the statistician must always keep apart the magnitude 
of the association and the reliability of this magnitude. We doubt the truth of this statement, but it 
cuts against his own argument at this point ! 
t The probable error of a coefficient of correlation for any frequency distribution is given by the 
formula : 
p.e. of r=-67449<r r = 
P3\ _ Pl3 ) ^ 
\/n W 2 T 2f>2o2>02 ' 4 i>20 2 ' ±Por PnPw PnPw) 
_ -67449;- \p 2 , , p 22 
In \ 
, Pit) . Pot 
+ ~, 5 T ■ 
(Pearson, Drapers' Research Memoirs, " General Theory of Skew Correlation," p. 20). 
The whole work therefore of finding <r r turns on evaluating p n , p^, P31, P13, P20 an d p m for a 
fourfold table : 
a 
b 
a + b 
c 
d 
c + d 
a + c 
b + d 
n 
when we suppose concentration at single points of the frequencies. 
We write ad-bc = e, (ad-bc)jn 2 = e' , 
{a + b)(c + d)(a + c){b + d) = q, and ql>i i = q'. 
Then we have r=e'jjq', 
,/c + d a + b\ /b + d a + c 
Pn = t 
P20 : 
(b + d) {a+c) 
p 22 = q +e 
(a + b) {c + d) 
P02= ^ ' 
#31 = e 
Write for brevity 
and 
Then 
And 
,(b + d)Z + (a + e) 3 
Pu = e 
c + d 
z ~n~' 
b + d 
n ' 
,{c + d)3 + {a + b)i 
mi 
, a + b 
a+c 
1111 
:— + — + — I + — 
m l mi n«i m 2 
f Jt±*E + «|±tf + (£±f + (a ±^ = n W - 6). 
a + c 
b+d a+b 
c + d 
2i»20P0 
P22 _., ,., , (l»l-»ll')(»t2-»l2') 
Pn 2 f 
7)22 lie', ,. , ,. 
-- + -,- {"h - % ) (™2 ~ »<2), 
1 'I 
PnPm PnPo2 « 
Biometrika ix 
22 
