267 
OF ERRORS OF JUDGMENT AND ON THE PERSONAL EQUATION. 
Further, 
Sx'gSu = SxgOii — 
and 
^x'pr jfi — rrx^rr '2^« > 
whence 
^ UX'a - 
0-,- 
I q 
Thus (xiv.) gives us (xv.) and (xvi.) The numerical values obtained 
were, aj, being in half-inch units :— 
Table XI. 
Quantity. 
1. 2. 
3. 
rux 
•9640 
•9514 
•9613 
^x' 
•1306 
•1635 
•1402 
'I'ux' 
- -0186 ±-0302 
+-1465 ±-0295 
+ -0851 ±-0299 
Now is the correlation between the absolute error made l)y the^th observer and 
the length of the line bisected, and we see at once that, contrary to our d priori assump¬ 
tion, there is little relationship between the amount of error and the length of the line 
bisected. Dr. Lee even makes a larger absolute error for small than for large lines, but 
her correlation is below its probable error in value, and we can only conclude that 
the length of the line between the limits taken for it in the experiments is quite 
immaterial to her judgment of its midpoint. There is a small correlation between 
Mr. Yule’s error and the length of the line, his error increasing if the line be longer. 
I am the only one of the three experimenters whose judgment of the midpoint of a line 
is considerably influenced by its length, but even in my case the result is of a totally 
different order from what we d priori had anticipated, for we had supposed the error 
would be almost directly proportional to the length of the line dealt with. 
Clearly, in correlating the judgments of (l) and (2) or of (l) and (3) we should 
have done better to take absolute displacements of the midpoint, rather than the 
proportions these bear to the length of the line. Accordingly I proceeded to deduce 
formulae for finding the correlations between the absolute displacement errors. 
Since 
V 't 
/ 
-V* - rgy - - 
2 > 
we have, by multiplying out and summing 
OC q — 'X'q 
x\ = Xr^ 
(Xx' CT* V '} x' X' ' ' — T X ^ 
•*•0 •*<} -t-D* 
Sx'^ — Sxp — 
5 
(r'^'Xurr.xP 
rn-' t' — 
2 U^X^ U 
^X',, ^X'p 
2 M 2 
Or 
(xvii.). 
