Karl Pearson 
301 
But S 
and accordingly : S |) = - .V^ + etc. 
We have then 
1 fy 
Again turning back to (xxi) : 
(1 - 7^2) _ Z = r ry' + f2 (^'2 _ ] )\ + 1^2 (1 + y2) (] + y'2Y 
Square this, multiply by (xxii) and nJN, sum and note that sum of odd powers of 
y' go oiit. We find 
Substituting (xxiii) and (xxiv) in (xviii) we deduce 
2 _ (1 - 77^)2 r .,2 „_ _ „2 
N 
(1 - 7,2)2 
^ — + rr^^2 + mi + terms ni t,'- 
1 + y2)2 (1 + y2)2 iV2z2 
^2 z! 
iV2z2 ' iV (1 + y2^2 
+ -AT n I ..2X2 + terms in t,^ 
, , , ^ „ -67449 (1 - 7,2) ^ 21^2 y' 
Probable Error of 77 = j= ^ ] J^-v. + ttt ti— oxo 
ViV ^2 iV (1 + y2)2 
X (1 + terms in 7,2 and higher powers) (xxv). 
This expression is of value for two reasons. First it indicates that the probable 
error of rj found for non-associated variates is 
67449 [ 1.-^1.2 y2 
VF [N'^z^ iV (1 + y2)2 
XXVI 
And secondly (xxv) may give a good approximate value of the probable error of 
7j, when we neglect the last factor altogether, say for values of tj under -4 or -5. 
Thus in the illustration given above 
y2 ]i_ f-548,861 ^_ ^_^2332 '^^^^^ 
NH^ iV(l + y2)2j I -31170 ' (1-31170)2 
= 1-4208. 
We have therefore to multiply the probable error as given by the product-moment 
method, i.e. in our case -0147 by 1-4208, giving -0209, in fair accordance with the 
more accurate value -0226, it being remembered that a probable error is rarely 
worth more than one or two significant figures. 
