358 On Hierarchical Order among Correlation Coefficients 
If we let a'l be the mental test " Rearranged Letters," 
„ ,, " Missing Digits," 
a ,, ,, „ " Analogies," - 
the values there found were 
r,.,„ = ()-61. 
Then by the above formula the correlation of the errors of these two coefficients 
depends chiefly upon rj._,,,,, whose measui'ed value is 0'63. Using the full formula, 
and employing the measured values in defiiult of the true ones, the correlation 
between r,.j„ and r^,a turns out to be "47. It is therefore (to an extent indicated 
by this value) probable that they ai'e either both too large or both too small. 
The same argument holds, in varying degrees, for the other correlations all over 
Mr Wyatt's table, which are all positive. They have a tendency to be either 
all too large or all too small : in other words, the es tend to be all of the same 
sign. The relationship between the correlation coefficients of a column, and their 
errors, can therefore be summed up in the following table, in which the symbol 
I e 1 denotes the magnitude of e regardless of sign. 
TABLE I, 
r 
\e\ 
P 
or f 
or p'e 
large 
small 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
V 
A 
+ 
+ 
+ 
+ 
+ 
+ 
small 
large 
I 
+ 
+ 
or + 
The first column shows the true correlations r' arranged in order of magnitude. 
The second column expresses the fact that the sampling errors on any occasion 
will probably be arranged in the reverse order of magnitude, disregarding their 
signs. The third column shows the correlation coefficients measured from their 
mean. The upper p"s are then positive, and the lower negative, and also, what is 
not shown in the table, the absolute values increase upwards and downwards from 
the point where the signs change. The fourth (double) column shows the probable 
arrangement of the signs of the quantities e. If the e's are all tending to be 
positive, then the left-hand member of the double column gives the arrangement, 
while if the e's all tend to be negative, the other member of the double column 
does so. As shown in the last (double) column, therefore, the quantities p'e tend 
either to be nearly all negative or nearly all positive. For a very small sample 
the signs of p'e will no doubt be quite irregularly arranged. But with such a 
