358 An Objective Studij of Mathematical Intelligence 
The discrepancies between mark-values and grade- (or adjusted rank) values 
are due to deviations from normal distribution*. This is empirically obvious 
from the fact that the discrepancies are larger the larger the deviations from 
normal distribution. This is also the case with the following additional coefficients, 
calculated by only two of the methods : — 
rJ-^ [marks] 
»- = 2 cos ^(l-7^)-l [ranks] 
Arith. G 
•78 +-03 
•75 
Alg. Gt 
•71 + ^04 
•57 
Geoiu. G 
•67+^04 
•60 
AC 
•78±^03 
•81 
AD 
•65 + ^05 
•60 
FG 
•64+ •OS 
•63 
BC 
•60+ •OS 
•68 
CI 
•55+^06 
•57 
CF 
•51 + •oe 
•50 
AB 
•49 + •oe 
•65 
BD 
•37 + -07 
•29 
Thirty-three other coefficients were worked out by the formula 
r=2cos|(l-i?)-l 
only. They are as follows : — 
A 
•81 
HI 
•92 
AF 
•46 
B 
•55 
EF 
•82 
DE 
•46 
C 
•94 
EG 
•76 
AE 
•44 
D 
•82 
AH 
•61 
DI 
•43 
C 
•54 
CH 
•57 
AG 
•41 
D 
•43 
EI 
■57 
DH 
•40 
E 
•91 
FH 
•55 
BF 
•28 
F 
•80 
GH 
•53 
BH 
•26 
C 
•59 
DF 
•49 
BE 
•23 
D 
•41 
AT 
•47 
BI 
•22 
H 
•96 
CE 
•47 
I 
•93 
All the above coefficients are "crude," i.e. uncorrected for the effect of extraneous 
conditions. In the present case the disturbing conditions are principally difference 
of age and difference of class or form. To correct for the first, coefficients were 
evaluated between age and each of the other twelve series. For ranks 
r = 2cos|(l-ii)-lj 
* Considering the smallness of the sample, and therefore the largeness of the random deviations from 
possible normality, the two formulae lead to results in strikingly close accordance and suggest normal 
distribution as being the rule for the psychical characters observed. 
t The exceptionally large discrepancy here corresponds with the fact that the characters compared 
have the two most skew distributions of the series, their distributions being: Alg. 26, 16, 19, 9^5, 6^5, 
4, 1, 1 (a J-curve). G 30, 26, 12, 4, '2, 9 (a U-curve). Unit of grouping, 7 marks. 
