300 On the Probable Error of Biserial -q 
For comparison we will investigate tlie value as given by the less exact formula 
(xix) : 
•805,191 
1689 X (1-31170)2 
•051,338 + -31170 (-102,676 + -423,327) 
+ 9-739,422 (-548,861 + -839,598 - -995,062)] 
= -000,277,077 x 4-046,752 = -001,121,26. 
Hence a, = -03349, 
and Probable Error = -0226. 
Thus for all practical purposes (xix) gives as good a result as the more lengthy 
formula (xiv). Had the value r = -3204 been computed by the product moment 
method for a population of 1689 the probable error would have been -0147. Thus 
the accuracies of the product-moment and the biserial t] methods appear to be 
about roughly in the ratio of 2 to 3 for this case. 
We will now turn back to the formula (xviii) and endeavour to express the 
quantities in curled brackets in powers of r ov -q. We have 
1 1 o 1 i t-y -ry')^ 
i -27s- i "2 1-^2 
a/277 V27r 
V2tt 
Hence 
y 
z 
y 
z 
y 
{1 - ryy' + ^r^ (y^/y'^ + tj"^ + y^)} 
ry' + r2 {y'^) + ^r^ {1 + y^) (1 + y'^)\ (xxi). 
Again — = -7= e " dx, 
V2TT'ys 
and ys = y ~ + ~2 *'^y- 
Thus %^ = ^+ 1 e'^^'dx. 
Take x = y — x', so that x' will be small, thus 
N 
0 
[1 + yx' + |y2x'2) (1 - la;'2) di' 
V 
Similarly : 
1 J- z {ry' - lyr' + If-yy'"-) + etc. 
N 
iV2 
z (ry' — lyr^ + i^^y,'/'^) — etc., 
ry' - z^r^y'^ + Z ^ hyr^ (1 - y'^) + etc....(xxii) 
