A "Unified" Intelligence Scale- EloIQ

copyright 1998 Bill McGaugh

In Gift of Fire #94 (the journal of the Prometheus Society), Robert Dick wrote a short piece suggesting that the Elo system, used in rating chess players, might be used as an intellectual rating system. This comment motivated me to think about the actual implementation of such a system. The following is a description of my first attempts at using the principles of Elo to develop a novel intelligence scale.

Darryl Miyaguchi recently supplied item analysis data for the Mega test on his Difficult IQ tests site. I found this information very interesting and useful in developing a new intelligence scale.

I started with the data for the top 100 scorers on the Mega test. I made a table of the number correct for each problem on the test. I decided to assign a value of 2500 to the typical top 100 person. Then I took the 12 most difficult items, and for each item, I looked up the percent that got the problem wrong on Elo's Percentage Expectancy Table (for example, item #48 was answered incorrectly by 48% of the top 100, and that implies a rating difference of 14 points). Since the average top 100 person is defined as 2500, problem #48 is assigned a rating of 2486. This assignment of Elo ratings to the problems themselves is the primary conceptual innovation in my system.

After finding the values for each of the 12 most difficult problems, I was ready to assign Elo ratings to people. The 12 problems had an average rating of 2483. So, if person #1 on Hoeflin's list got 11 out of the 12 problems, I treated him as though he had won 11 out of 12 chess games from an average rating pool of 2483, and his performance rating was 2884. I continued down the list assigning ratings to each of the top twelve people on the Mega list.

Person EloIQ
1 2884
2 2756
3 2676
4 2676
5 2608
6 2884
7 2540
8 2540
9 2608
10 2608
11 2540
12 2676

The beauty of the system is that we can compare individuals with problems that have been assigned Elo ratings and make predictions about the probability that an individual with X rating can solve problem Y. For example, problem #36 (interpenetrating cubes) has a difficulty rating of 2649. Person #1 has a EloIQ rating of 2884, a difference of 235. From Elo's tables, this means that #1 has about 79% chance of solving the problem. I find this to be much more useful and interesting than a statement like: person A has an IQ of 150, which is only found in one person out of a thousand.

Continuing to work with the Mega data I was able to assign the following tentative EloIQ ratings to various IQ/Mega levels:

Mega IQ EloIQ
---- 100 1282
9 132 1832
24 150 2144
36 164 2357
44 180 2666

The linear approximation eloiq=1282+(IQ-100)*17.3 predicts eloiq from IQ. Converting the other way, we have IQ= (EloIQ-1282)/17.3 +100.

It is interesting to note that even though my initial choice of 2500 was arbitrary (we can shift the entire scale as we choose), I ended up with a scale that that mirrors chess ratings rather closely. There is a problem, however, in mapping chess ratings to IQ directly, since a study of German players with an average chess Elo of 2300 found that their IQ's averaged only 110 IQ. The other problem is that study and two others have found that the correlation of chess ratings to IQ is close to zero. A study is needed that includes only people with several years of experience at tournament chess, across all chess rating levels. Using the EloIQ method, we could present the players with a set of rated problems and study the correspondence between their chess Elo and their EloIQ.

A strange side effect of the EloIQ system is that it may provide a method of directly comparing deviation IQ's with chronological ratio IQ's. .
In 1951, Vernon Sare studied the relationship between deviation IQ's and ratio IQ's. Darryl Miyaguchi reviewed the highlights at the Brain Board. If you take the EloIQ and divide it by the EloIQ for IQ 100 (1282) and the result is a number that is very similar to Sare's predicted ratio IQ. For example, a deviation IQ of 180 (2666 EloIQ) would be equal to about a 208 ratio IQ. Sare's prediction is that it is a bit over 210. Coincidence? This relationship, and similar connections with chronometric scales, will be the subject of future study.

Prometheans have already supplied many ideas about the potential applications of the EloIQ system. Problems can be normed one at a time and used for society admission. Robert Dick proposed the creation of an ongoing intellectual tournament, where members and subscribers would try problems and their EloIQ ratings would rise and fall. As is the case with chess, it will take some work to maintain stability in the rating system.

With the legal challenges to unlicensed IQ testing, the EloIQ system (which could be called anything we choose) allows Hi-IQ societies to continue to exist and accept applicants based on their ability to solve problems at a level equal to the g levels of the present and past members.

This system seems to be a step toward a scale unifying deviation IQ's, chronological IQ's, and, possibly ratings in intellectual games, such as go and chess. The "grand unification" will come with the inclusion of chronometric measurements in the model. At this point, I can only make preliminary guesses about the mapping of chronometric measurements onto the EloIQ scale.

All of the numbers in this article are very preliminary. Future studies will refine the problem difficulties and ratings.

Some Mega Test Problem Difficulty Ratings:
Problem Difficulty
36 2649
20 2649
21 2625
22 2625
48 2486
23 2420
24 2413
30 2405
7 2405
33 2390
41 2367
47 2367

Some Test for Genius Problem Difficulty Ratings:
Problem Difficulty
20 2510
44 2510
27 2493
29 2493
14 2476
23 2441
30 2441
9 2407
17 2407

Extra work was required to derive accurate difficulty ratings for the problems that are very simple for the top 100. The method I used was to work my way down through subsets of the sample and re-calculate. Most of the problems on the Mega test have ratings of at least 1900, which is why people with IQ 132 (1832) only average 9 correct.

Sample Percentage Expectancy Table:
Difference Percentage
0 50%
36 55
72 60
110 65
149 70
193 75
240 80
296 85
366 90
470 95
677 99
>735 100