Jan 29, 2011

D&D and IQ

I first started playing RPGs about two decades ago with Advanced Dungeons and Dragons.  Back before 4th edition finally defaulted to a point-buy system for attributes, character attributes were determined by rolling six-sided dice (d6).  The basic and most scary method was to roll 3d6 and add the results, providing a total from 3 to 18.  The "heroic" method was to roll 4d6 and sum the highest three results, which still provided a total from 3 to 18, but with a higher mean.

It had generally been said in my gaming circles that Intelligence (INT) in D&D was the equivalent of an intelligence quotient (IQ) divided by 10.  Or INT * 10 = IQ.  The average IQ is 100, and the D&D manuals stated that the average INT was 10 (really 10.5 by rolling 3d6).  Heroes, rolling 4d6 and dropping the lowest, have a mode of 13 and a mean of 12.2446.  The whole INT distribution, however, does not fit as well as its mean onto the real IQ distribution.

The old Stanford Binet calculated IQ as 100 times the ratio between a child's age and the age of children with the same performance on average on particular tasks.  So, a 6-year old who performed as well as the average 9-year old would have an IQ of 150.  This method of determining IQ scores becomes less useful as children grow up, and makes no sense to apply to adults.  The newer Wechsler tests calculate IQ by taking the scores of everyone in an age group and putting them on a normal distribution with a mean of 100 and a standard deviation (SD) of 15.  This is a much more clear way to evaluate how a person functions intellectually compared to same-age peers.

If we take the distribution of 3d6 rolls and associate the probability of each sum with the prevalence of corresponding IQs in the real world, we can get a better impression of what a D&D INT score means in terms of IQ.  If we say that 3d6 simulates a normal distribution of intelligence, then here are the translations (rounded) of each INT score into an IQ score:

INT     IQ 
 3      57
 4      66
 5      72
 6      78
 7      83
 8      88
 9      93
10      98
11     102
12     107
13     112
14     117
15     122
16     128
17     134
18     143

The IQ scores in this table are calculated from the z-scores for the middles of each range of probability that maps onto each INT score.   For example, an INT of 3 and of 18 each have a 0.463 chance of occurring (1 in 216, which is 6^3), but I cannot use the z-score for .00463 to calculate IQ for an INT of 3 and work my way up because I would be left trying to use the z-score for 1 with the INT of 18, which is infinity standard deviations.  Also, I would end up using the z-score for .5 for an INT of 10 when it truly applies to an INT of 10.5.  What works is to use the z-scores for the midpoints between probabilities.  So, an IQ of 57 (2.84 standard deviations below the mean) corresponds to the z-score for .0023 (the midpoint between 1/216 and 0, since it is impossible to roll less than 3) with some rounding.  The probability of rolling at least a 9 is .375, and the probability of rolling at least a 10 is .5, the midpoint is .4375 for an INT of 10, which has a z-score of
about -.157.  100 (the mean) minus (.157 * 15 (the standard deviation)) equals about 97.645, so the IQ for an INT of 10 is 98 after rounding.  Each IQ score was calculated this way.

As you can see, multiplying INT by 10 does not give you your character's IQ.  Now, this does not address differences in intellectual ability by age group.  A stereotypical D&D party consists of characters emerging from adolescence into adulthood, roughly in the same stage of development, so it should be okay to use this scale.  A 12-year old with a 120 IQ still lacks much of the functioning of a 20-year old with a 100 IQ, and a 30-year old with a 100 IQ functions a little differently than a 50-year old with a 100 IQ.  Age tends to result in a larger vocabulary and body of knowledge with a slower information processing speed, but I may go deeper into how RPGs simulate the effects of aging on intelligence in later posts.

For that matter, using an overall IQ score obfuscates the differences in each person's separate areas of intellectual functioning.  "Intelligence" refers to many different abilities that we have, and a dozen people with the same IQ can each function differently from each other on a variety of tasks.  Few simulation systems break down intelligence into components, and that is also a topic I will return to in future posts.

Jan 28, 2011


The "Sim" in SimAntics is short for Simulation.  This blog will be a collection of my intellectual antics with simulation systems, specifically tabletop role-playing game systems.  I am combining my gamer nerdiness with my math nerdiness to analyze various RPG game mechanics.  I look at how well different game mechanics simulate real life, and also how functional or balanced they are for game play.  Some of this blog will read like a manual for munchkins and min-maxers, some like instructions for game designers, and some of it will read like a statistics textbook.