Editorial 
127 
As in the case of the discoveiy of the best vahie of the median from two 
bahmced pairs of grades, it will be seen again that there are two factors at work. 
When the pairs of grades are close to each other then the correlation is high in 
the errors of a, and we lose the advantage of duplicating our determinations of a. 
On the other hand when the grades are taken far apart the correlation is low and 
the gain in duplicating our determinations should be greater. But in this case 
one or other pair of grades must be far removed from the grade which gives the 
lowest probable error of any determination of a. 
Our second table shows us that we must push one pair of grades considerably 
out from the pair, and the other pair considerably in. The lowest value occurs 
at a little under 12 increase of inaccuracy above mean and moment values of cr. 
The pairs of grades are approximately 3^ and jL or As the latter gives very 
closely the proportions "6 and the following rule was tested. Take '6 times the 
value of a as found from the interval between the grades and "4 times the value 
of a as found from the \ grades (i.e. the second deciles from either end). The 
mean square error of cr is then found to be I'll 75 exactly. This is close to 
the minimum value obtainable by the use of two pairs of grades. 
Of course by using three pairs of grades we can still further increase the 
accuracy of grade methods, but the choice of the three best pairs would require 
lengthy investigations for their determination, and every increase of complexity in 
grade investigations reduces their already small advantage over mean and moment 
methods. We may illustrate three pairs on the supposition that ^'j,, and y-^- are 
suitable grade jiairs. 
We have a ^ 1 o = X 1 cr , + A. , <t , + X_o cr 0 
75u > TI ' lu ao .{IP 11 IT lu 1T7 
leading to 
'^J^, 1 Tlu "^.J, 11 ITT <^'-^ 
a « I 1 ? ' 2 u ;j u 1 i 1 II 
+ 2X a X , [8<r I Str 1 ] + 2X , X . [80- , Scr . ] + 2X 2 X 1 [8(7 e Scr 1 ]. 
Tsu 1 4 Jil 14-" 14 111 14 To-" ru Sij '- lu TTU-' 
Substituting the numerical values from Tables VII and VIII we have the following 
equation from which to find X , , X^i_, X^2_ subject to the condition 
Xi+Xi+X. =1: 
Ti u 1 T 1 II 
<7\ =XS 1-6780,6116 + XS l-5:333,8(]89 + XS 2-1614,8804 
A- IT, 1% " " 
+ 2X , X 1 1-0.501,2591 + 2X 1 X . -9102,7433 + 2X . X 1 -6233,9299. 
:ro 1 4 1 4 1 u T 0 3 o 
We determine 
X , = -38325, X 1 = -32446, X . = -29229 (41), 
leading to o-^ ^ ^ = 1-0798 ^|=^ (42). 
a u , IT > \1S 
V2iV 
