Statistics of Numbers

© Paul Cooijmans


This was a number series test; very unfortunately, criminals are publishing many number series from tests with the answers on the Internet without permission from the tests' authors. It is therefore easy to cheat on such tests, and unwise to accept the tests' scores for admission to I.Q. societies.

I regret this problem especially in the case of this particular test, because originally it was an integral part of my Test For Genius from 1995. After discovering the copyright violation I felt forced to violate the integrity of that extremely beautiful, profound and brilliant creation by removing the numerical subtest. It is sad, but apparently there are people whose dishonesty and desire to help other dishonest persons - cheaters - is so great that they completely disregard intellectual property and the natural necessity to keep test answers secret. Our contempt for them can never be deep enough.

Scores on Numbers as of 13 February 2023

Contents type: Numerical.   Period: 1995-2010

6 *
7 **
8 *******
9 **************
10 ****************************
11 ************
12 ************
13 *********
14 **********
15 ********
16 *****
17 ****
18 *
19 ******
20 *

Scores by males

n = 114

6 *
7 *
8 ******
9 **************
10 **************************
11 ***********
12 ************
13 ********
14 **********
15 ********
16 *****
17 ****
18 *
19 ******
20 *

Scores by females

n = 6

7 *
8 *
10 **
11 *
13 *

Correlation of Numbers with other mental ability tests

Test name n r
Association and Analogies (French)50.91
Analogies subtest of Long Test For Genius (French)50.90
De Laatste Test40.89
Short Test For Genius280.87
Daedalus Test50.87
Long Test For Genius (French)50.86
Association subtest of Long Test For Genius (French)50.79
The Bonsai Test - Revision 201650.78
Reflections In Peroxide50.77
Psychometric Qrosswords40.71
Sigma Test (Melão Hindemburg)140.71
Long Test For Genius (Netherlandic)120.69
Non-Verbal Cognitive Performance Examination (Xavier Jouve)110.69
Analogies subtest of Long Test For Genius (Netherlandic)120.69
Reason Behind Multiple-Choice80.68
Analogies #170.66
The Nemesis Test70.66
Problems In Gentle Slopes of the first degree40.66
Isis Test110.66
Chimera High Ability Riddle Test (Bill Bultas)60.64
Mega Test (Ronald K. Hoeflin)250.60
Titan Test (Ronald K. Hoeflin)150.59
Cooijmans Intelligence Test - Form 2110.57
KIT Intelligence Test - first attempts40.57
Space, Time, and Hyperspace540.57
Long Test For Genius260.57
Problems In Gentle Slopes of the third degree60.56
Reason Behind Multiple-Choice - Revision 200870.56
Reason - Revision 200870.55
916 Test (Laurent Dubois)40.55
The Final Test - Revision 201350.54
Numerical section of Test For Genius - Revision 201090.53
Association subtest of Long Test For Genius300.53
Unknown and miscellaneous tests520.51
Combined Numerical and Spatial sections of Test For Genius - Revision 201070.50
Spatial section of Test For Genius - Revision 2004160.49
Qoymans Multiple-Choice #3140.47
The Final Test250.47
Cooijmans Intelligence Test - Form 390.47
Test For Genius - Revision 2004130.43
Cooijmans Intelligence Test - Form 1190.41
Spatial Insight Test60.40
Analogies of Long Test For Genius300.39
Qoymans Multiple-Choice #570.38
Problems In Gentle Slopes of the second degree60.38
The Hammer Of Test-Hungry - Revision 201350.38
Lieshout International Mesospheric Intelligence Test130.37
Test of Shock and Awe70.35
Drenth number series50.32
Genius Association Test170.31
The Test To End All Tests130.30
Narcissus' last stand50.28
Tests by Kevin Langdon (aggregate)50.26
Qoymans Multiple-Choice #4200.26
Strict Logic Sequences Exam II (Jonathan Wai)70.25
Logima Strictica 36 (Robert Lato)140.25
Qoymans Multiple-Choice #190.24
Cartoons of Shock110.24
Test of the Beheaded Man50.24
A Paranoiac's Torture: Intelligence Test Utilizing Diabolic Exactitude40.23
Association subtest of Long Test For Genius (Netherlandic)120.22
Ultra Test (Ronald K. Hoeflin)60.22
The Sargasso Test120.22
Cooijmans On-Line Test - Two-barrelled version60.19
Cattell Culture Fair190.18
Culture Fair Numerical Spatial Examination - Final version (Etienne Forsström)130.18
Raven's Advanced Progressive Matrices (I.Q.)120.17
Sequentia Numerica Form I (Alexander Herkner)50.17
Bonsai Test130.17
W-87 (International Society for Philosophical Enquiry)70.16
Wechsler Adult Intelligence Scales90.15
Strict Logic Sequences Exam I (Jonathan Wai)140.13
Associative LIMIT80.08
Raven's Advanced Progressive Matrices (raw)110.08
Hoeflin Power Test (Ronald K. Hoeflin)6-0.06
Verbal section of Test For Genius - Revision 200415-0.10
Tests by Greg Grove (aggregate)11-0.12
Test For Genius - Revision 20106-0.14
Strict Logic Spatial Exam 48 (Jonathan Wai)4-0.14
International High IQ Society tests (aggregate)4-0.18
Graduate Record Examination5-0.44
Cooijmans Intelligence Test - Form 44-0.53
Tests by Nicolas Elenas (aggregate)4-0.56
Test of Inductive Reasoning / J.C.T.I. (Xavier Jouve)6-0.59
Encephalist - R (Xavier Jouve)5-0.68
Scholastic Aptitude Test (old)4-0.70
Griffioen I.Q. Test (Rijk Griffioen)4-0.92
Epiq Tests (aggregate)5-0.94

Weighted average of correlations: 0.399 (N = 980)

Estimated g factor loading: 0.63

Ranking in above table is based on the unrounded correlations. All available data is present in this table, no tests are left out except for those with less than 4 score pairs. All known pairs are used, including possible floor/ceiling scores or outliers.

Estimated loadings of Numbers on particular item types

These are estimated g factor loadings, but against homogeneous tests (containing only particular item types) as opposed to non-compound heterogeneous tests. Although tending to surprise the lay person, it is not uncommon for tests to have high loadings on item types they do not actually contain themselves. Such loadings reflect the empirical fact that most tests for mental abilities measure primarily g, regardless of their contents; that the major part of test score variance is caused by g, and only a minor part by factors germane to particular item types. It is of key importance to understand that this is a fact of nature, a natural phenomenon, and not something that was built into the tests by the test constructors.

Typeng loading of Numbers on that type

N = 715

Compound tests have been left out of this table to avoid overlap.

Balanced g loading = 0.67

National medians for Numbers

Country n median score

For reasons of privacy, only countries with 3 or more candidates are included in this table. Ranking is based on the medians, and then alphabetic.

Correlation with national I.Q.'s of Numbers

Correlation of this test with national average I.Q.'s published by Lynn and Vanhanen:

Correlation of Numbers with personal details

Personalia n r
PSIA System factor - Revision 200740.97
PSIA Rare - Revision 200740.85
PSIA Cold - Revision 200740.71
Observed associative horizon190.47
PSIA True - Revision 200740.36
PSIA Ethics factor - Revision 200740.31
Gifted Adult's Inventory of Aspergerisms110.30
Observed behaviour260.28
PSIA Rational - Revision 200740.26
PSIA Introverted - Revision 200740.23
PSIA Antisocial - Revision 200740.20
PSIA Orderly - Revision 200740.14
Year of birth1080.12
PSIA Just - Revision 200740.11
Educational level570.10
Mother's educational level530.04
Father's educational level530.03
PSIA Neurotic - Revision 200740.02
PSIA Cruel - Revision 20074-0.12
PSIA Deviance factor - Revision 20074-0.18
Disorders (own)54-0.18
Disorders (parents and siblings)53-0.23
PSIA Extreme - Revision 20074-0.26
Cooijmans Inventory of Neo-Marxist Attitudes4-0.38
PSIA Aspergoid - Revision 20074-0.44

Correlation with personal details of Numbers - within females

Personalia n r
Observed behaviour40.87
Year of birth60.65

Correlation with personal details of Numbers - within males

Personalia n r
Observed associative horizon160.58
Gifted Adult's Inventory of Aspergerisms90.39
Observed behaviour220.25
Year of birth1020.10
Educational level540.09
Mother's educational level510.04
Father's educational level510.02
Disorders (own)51-0.17
Disorders (parents and siblings)51-0.26
Cooijmans Inventory of Neo-Marxist Attitudes4-0.38

Estimated g factor loadings for restricted ranges

In parentheses the number of score pairs on which that estimated g factor loading is based. The goal of this is to verify the hypothesis that g becomes less important, accounts for a smaller proportion of the variance, at higher I.Q. levels. The mere fact of restricting the range like this also depresses the g loading compared to computing it over the test's full range, so it would be normal for these values to be lower than the test's full-range g loading.

Below 1st quartile0.36 (412)
Below median0.32 (537)
Above median0.55 (597)
Above 3rd quartile-0.47 (220)



Scores by age

Age class n Median score
80 to 8417.0
70 to 74110.0
60 to 64212.5
55 to 59511.0
50 to 54910.0
45 to 49512.0
40 to 441310.0
35 to 391413.0
30 to 341112.0
25 to 291811.5
22 to 241010.5
20 or 21814.5
18 or 1949.5

N = 106

Scores by age - within females

Age class n Median raw
40 to 4438.0
35 to 39212.0
22 to 24110.0

N = 6

Scores by age - within males

Age class n Median raw
80 to 8417.0
70 to 74110.0
60 to 64212.5
55 to 59511.0
50 to 54910.0
45 to 49512.0
40 to 441010.0
35 to 391213.5
30 to 341112.0
25 to 291811.5
22 to 24911.0
20 or 21814.5
18 or 1949.5

N = 100

Scores by year taken

Year taken n median score

ryear taken × median score = 0.41 (N = 118)

Robustness and overall test quality

Item analysis

Item statistics are not published as that would help candidates. To detect bad items, answers and comments from candidates are studied, as well as, for each problem, the correlation with total score on the remaining problems (item-rest correlation) and the proportion of candidates getting it wrong (hardness of the item). Possible bad items are revised, replaced, or removed, possibly resulting in a revised version of the test.