24 On the Probable Error of a Coefficient of Correlation 
We have (6 + d)/N = |(1 - a,), (a + c)fN = h (1 + a,), 
(a + b)/N = J (1 + «,), (c + d)/N =h(l- a,), 
a + d (a + b) (a + c) ac? — 6c (c + c?) (6 + cZ) ad — 6c 
~n~ = iv^ ~ + m + ~ ~W* + ~Jv*~ 
= i (1 + «0 (1 + «,) + 1 (1 - a,) (1 - eO = HI + «M 
since ad — 6c = 0 in the original population. 
Similarly : = ' J ^ — a i a .;)- 
a6 — cc2 a (N — a — c - d) — ca 1 
^V 2 ~ = TP - 
a (a + c) (a + d) 
= + 0(1 + «i) - i(i + «i) (i + «iO 
= K(i-«i 2 ), 
and similarly : ~~~m>~ = i°i (1 — a a 2 )- 
Hence substituting in (i) 
"G7449 
Probable error of r = ^jj^ [to (1 ~ W) + iW ( 1 ~ «i 2 ) + t<j«i 2 (1 ~ a *) 
-^(i-^^-KCi-rf 
•67449 
This can also be put in the form : 
-q 1 -ii , -67449 /(a + 6)(a + c)(d + 6) (d + c) ,. , 
Probable error ot r = —= a / — ~ — (rv). 
This is the jorobable error of r of a fourfold table when the real value of r is zero. 
Now as (ii) and (iv) give the reducing factors for the two cases (a) when h and 
k are both zero but r has any value and (6) when h and k have any values but r is 
zero, it occurred to me that the combined product of the two would give good 
results for a considerable range of values of h and k and r. We have to note that 
(iv) for h and k zero becomes 
•67449 7T 
Hence we take as our formula : 
Probable error of r 
= '67449 /j— ^ / /sin-irV ^( 1+^)^(1 -a.) Vj(l + a 2 ) j (1 - «,) 
ViV V V 90° / if if 
