Karl Pearson 
23 
and h and k have their usual meaning denned by the integrals 
h 
e ~ * z ~dz = ^oti , say ; 
(a + c)-(b + d) _ 1 C\-y> dz 
2N V27T-/0 
(a + b)-( c + d) = 1 i* 
2N V2 
e i z ~dz = \a 2 , say. 
IT J 0 
Let H = e~ -J 1 ' , K = e~i k " as usual. 
V2tt V2tt 
Now the formula (i) above for the probable error of r is admittedly laborious 
in use. I have tried in many ways, while retaining its full accuracy, to throw it 
into a form involving less laborious calculations ; I have not succeeded, however, 
in achieving any sensible reduction in its complexity, as long as I maintain its 
complete generality. 
Although many hundred fourfold tables have now been published, many of which 
give such small correlations that their true significance can only be settled by 
a knowledge of their probable errors, yet I find only 40 to 50 probable errors have 
so far been determined. This matter seems so regrettable that I have sought for 
a fairly easy method of determining a closely empirical expression for the probable 
error of r which is likely to be of service, and can be adapted easily to tables. 
I consider first two extreme cases. If h and k are both zero, or the fourfold 
division at the mean, then ^ = ^, = 0 *, 
Probable error of r 
_ -67449 27rVl-r 3 \ {a + d) (b + c)| * _ "67449 Vl - r 2 nr [16a6|* 
" ( 4iV 2 J " ~ 2 \W\ ' 
since in this case a = d, and b = c. 
But for a division at the mean by Sheppard's Theorem 
b . fir nrb 
r = cos 7r r = sin T 
a + b V- a + b 
or (sin" 1 r)/^-rr = (a - b)/(a + b). 
/sin -1 r\ 2 4a6 IQab 
Hence 1 — 
Substituting we have : 
^7T J (a + bf W 
r. u ui c '67449 77 / ,sin _1 r\ 2 .... 
rrobable error ol r = - . VI — r- \ 1 — - (n), 
*IN 2 V V 90° / V ; 
if the angle of the inverse sine be read in degrees. 
Again if r = 0, the probable error of r may be obtained from (i) whatever the 
values of h and k. For in this case 
ad — bc = 0, ^1 = ^1, ^2=^01.2, x<> = HK. 
* Phil. Trans. Vol. 192 A, p. Ill and Vol. 195 A, p. 7. 
