242 
PROFESSOE K. PEARSON ON THE MATHEMATICAL THEORY 
correlated. Independence of absolute judgments connotes correlation of relative 
judgments. This is, of course, an instance of what I have termed “spurious” 
correlation, but it is none the less important that it should not be overlooked. When 
we cannot form absolute judgments, but refer our observations to a special observer as 
standard, then the observations so reduced of two independent observers will certainly 
be correlated. I am not aware that attention has hitherto heen paid to this point 
when the observations of different observers relative to a standard man have been 
combined. 
One result of the actual correlation of independent judgments is that the values 
experimentally determined for the p’s are not those given by (viii.). A genuine 
correlation is superposed on the sjDurious correlation, and the total correlation 
observed may be greater or less than the values indicated in (viii.). 
(4.) New Theory of Errors of Ohservatio'n. 
Let us suppose that the correlations r’ 33 , r^^, are not zero, then, provided we 
calculate the standard deviations of the absolute and relative judgments, we can find 
at once these correlations. We have 
— 
31 — 
2 I 2 
^02 • ^03 ^23 
-^oa®’o3 
0,0 o 
01 
2crnncr„ 
'31 ^ I 
' 03*^ 01 
0,0 o 
^01 I ^n3~ ^ia~ 
■^'^10^03 
(ix.) 
We are no longer able to find the absolute variabilities from the relative variabilities, 
and we require direct experiments in which the errors of absolute judgment are 
knoATO in order to determine the correlations. 
Turning now to the correlations between relative judgments, we easily deduce 
from (iii.) 
P-iy \i 
T ^’l3®'oi°’o2 '^’si^OS^Ol '*’32®’()3°’02 
+ aod - 2 
^03^01' 
'3l)\/( 
O', 
03 
+ <^ 02 “ 
2o'(j30‘(,p33) 
^ I 
O'.,," + cr. 
32 
'12 
*"‘^3l‘^32 
since 
cr 
31 
0 I O ;■ 
— O'os + o-oT ~ ‘ 
cToaCT, 
oH'so 
and similar relations hold. 
