264 
PEOFESSOE K. PEAESON ON THE MATHEMATICAL THEOEY 
The bright-line experiments show in a perfectly direct and simple manner that 
the correlation of absolute judgments is not wholly due to some external source 
influencing all observers in the same way, but is the result of a common element in 
the personalities of two observers. They further demonstrate the extreme difficulty 
in actual observations of separating without experiments acl hoc this psychological 
from the spurious correlation. These are precisely the points they were designed to 
elucidate. 
(ii.) The Bisection Ex])eriments. 
I have indicated that it was the correlation in judgments of independent observers 
in the case of bisection that led to the second series or bright-line experiments. 
After these had demonstrated that the correlation of judgments was not wholly 
spurious correlation, it seemed desirable to reconsider the bisection experiments with 
a view to analysing more fully the character of the correlation exhibited by them. 
The reader will remember that the error in judgment in their case was taken to 
be the displacement to the right of the true midpoint measured as a fraction of 
the total length of the line bisected. The following are the values of the corre¬ 
lations between the absolute judgments and between the relative judgments thus 
measured :— 
Table VII. 
ros = -3627 ± '0262 
Pi) 23 = ‘5615 ± -0207 
rsi = -llSg ± -0298 
p'2^ 31 = ’4980 + *0227 
1 
ri2 = -2053 ± -0289 
PS) 12 = '4379 + '0244 
Thus in every case the correlation has a quite sensible value. 
I have pointed out that the absolute displacement of the midpoint by the experi¬ 
menter was divided originally by the length of the line, because d priori we supposed 
that errors of bisection would be proportional to the length of the line l:)isected. But 
that when I had more fully realised the meaning of spurious correlation I saw that the 
whole of the above correlations might be really sjDurious in character, for they were the 
correlations of ratios having the same denominator. The expei’iments were accord¬ 
ingly put on one side until the bright-line experiments were concluded. It then 
seemed desirable to determine the correlations between the absolute displacements 
of the midpoints, and to find the magnitude of the correlation between the lengths 
of the lines experimented on and the errors made in their bisection. The labour of 
reducing again all the data would be excessive, and a very little consideration showed 
me that it was really unnecessary, if Ave kneAv the variation and distribution of the 
lengtlis of the lines bisected. Let u stand for the length of any one of the bisected 
lines, Avhich as we have seen Avere a random sample. Then Ave haAm the folioAA'ing 
distrihution :— 
