# Matching Dungeons and Dragons statistics to reality

The rules of Dungeons and Dragons use concrete numeric values to describe how strong, agile, rugged, smart, wise and charming a character is; it uses these, combined with the results of die-rolls and concrete numeric values for how difficult assorted tasks are, to decide whether a character succeeds or fails in assorted tasks. There are all sorts of silly results that can arise from applying the rules but they're simple enough to let the game flow dynamically, rather than getting bogged down in working out what the rules imply, yet faithful enough to reality that for the most part things the players' characters actually tend to get up to tend to work out reasonably realistically.

Naturally enough, players are wont to wonder how they themselves would stack up, in this system. If nothing else, it's a rôle-playing game, so players are obliged – like method actors – to think in character; to do that, they need to understand their character's vital statistics, for which it makes most sense to try to relate them to people the player knows. There's some diversity of opinion as to how to actually model reality using the game's numbers. One fairly well respected, if controversial, source is the Alexandrian; and there's at least one web-site at which, by answering a bunch of questions about yourself, you can get one author's opinions as to how you stack up (here are my results). I don't pretend to know the game well enough to speak decisively about that, although I'll necessarily take (hopefully fairly anodyne) positions on some details; but I know a thing or two about statistics, that I suspect can illuminate the discussion.

## Assumptions and Conventions

The rules overtly state that an ability score of 10 is a typical value for a normal person, so I'll take that as given. I'll use the orthodox abbreviations Str for strength, Dex for dexterity, Con for constitution, Int for intelligence, Wis for wisdom and Cha for charisma, as ability scores; for the associated modifiers, I'll use the lower-case form, str = (Str−10)/2, dex = (Dex−10)/2, con = (Con−10)/2, int = (Int−10)/2, wis = (Wis−10)/2 and cha = (Cha−10)/2, with fractional answers rounded towards zero.

Like the Alexandrian, I'll not take it as given that if it happens in the game, it happens in reality – after all, magic happens in the game – so the fact that the game provides for players to attain levels above five (the highest the Alexandrian countenances anybody in reality being), all the way up to twenty and (in the epic extensions) beyond, doesn't mean there necessarily are people of such high level in reality; nor does the fact that the game provides for ability scores ranging from three to eighteen (and beyond) imply that this is necessarily the range of actual human scores in the abilities. Instead, in so far as I try to model reality myself, I'll look at what rules say about what practical consequences one gets out of having a given ability score; comparing this to practical experience in the real world, as far as possible, seems the most realistic way of deciding what one's ability scores would be, by the rules. Except in so far as I have clear contrary evidence, I'll also assume that the mapping between real-world statistics for ability scores is the same from one ability score to another; if I find contrary evidence, I'll be less willing to use different mappings among body-based scores (Str, Dex and Con), or among the mind-based scores (Int, Wis, Cha), than between these two groupings.

I'll use the conventional notation of Dungeons and Dragons for denoting die rolls: a number, n, followed by the letter d followed by another number, m, denotes the result of adding up n independent samples from a uniform random variate whose possible values are the whole numbers 1 through m; if n is omitted, the shortened denotation represents a die with m faces (numbered 1 through m) – the usual way to obtain samples from such a uniform random variate is indeed to roll a die with the requisite number of faces. Thus a d6 is a six-sided die, with faces numbered one through six; 1d6 means the number shown by a d6, when rolled; 3d6 means the result of adding up three such, from separate rolls (or from rolls of separate dice). The platonic solids provide for dice with 4, 6, 8, 12 and 20 sides. It is conventional also to use a ten-faced die (connect a planar regular decagon's vertices to two points equidistant from its centroid along the line, through this centroid, perpendicular to the decagon's plane; fill in the resulting wire-frame and you've got roughly the usual shape) to generate digits and a second, with faces numbered 00, 10, … 90, in combination with this to generate random numbers from 1 to 100 (rolling 00 on the latter with 0 on the former is read as 100); this combination is denoted d%, effectively using % as a short-hand for 100. For other numbers of faces, a suitable uniform random variate can be obtained by appropriately either scaling or filtering the given dice. A d7 can be simulated by rolling a d8 and re-rolling it as long as it comes up 8; the first other result it gives is indeed uniformly distributed over 1 through 7, provided the d8 was fair. The same can simulate a d13 from a d20 by re-rolling on any result greater than 13. For factors of the number of faces of a real die, one can use arithmetic: a d5 can be obtained from a d10 either by halving the result and rounding up (so 1 and 2 are 1, 7 and 8 are 4, etc.) or by reducing modulo 5 (i.e. subtract five if the result is greater than five) – but be sure to say, before rolling, which way the outcome is to be interpreted ! For d2, one could flip a coin or apply arithmetic to any of the platonic dice.

## Simple misconceptions

One common assumption is that Int corresponds to IQ divided by ten. Certainly, IQ is defined to have a mean of 100, so dividing it by ten shall give something with roughly the right mean; however, it will also (again because of how IQ is defined) have a standard deviation of 1.5, which may be unreasonably narrow. For example, common ways of generating ability scores for characters (albeit this isn't necessarily indicative of anything matching reality, either: see below) tend to produce a standard deviation of about three; so IQ/5 −10 might produce a more appropriate distribution for Int values. In any case, IQ (ignoring diverse arguments about whether it actually lives up to what its designers intended) aims to be an objective measure of innate intellectual potential – whereas DnD's Int score is deliberately a gestalt measure of how able the individual is to actually perform an assortment of intelligence-related tasks, along with a measure of how much the character knows; and it can be improved by training – it is not even trying to measure the same thing as IQ. Int doesn't care about your potential, it cares about how well you have actually learned to use your grey matter.

As noted above, I don't presume that the standard game mechanics for generating character ability scores necessarily say anything about what ability scores real people would have: but it is at least reasonable to entertain the idea that the game designers intended their rules for generating non-player characters' ability scores to produce a background population, for players to interract with, whose abilities would roughly match that for real-world general populations. However, for such an approach, it is important to distinguish between the game designers' guidance on how to generate ability scores for non-player characters (representing a population of normal people) and the corresponding guidance for player characters. The latter tend to be biassed in favour of producing characters with better than average abilities (i.e. scores above ten) – whether justified pragmatically as a way to give the players characters with some chance of success (which is more fun to play) or justified by saying that the player characters are the protagonists of a story, so of course they're out of the ordinary. Only relatively sophisticated cultures bother to tell the tales of ordinary people …

## Basic statistics

Given that some do use the character-generation rules as guidance to the range of ability scores to expect the real-world population to have, let's start by looking at the statistics for some such rules (in simplified form), if only to get a feel for what we're dealing with. I'll start with the least appropriate, to get it out of the way.

One common way of determining the ability scores of a player's new character is, for each score, to role four d6 and use the largest sum of any three of the results. This has a skew distribution: we preferentially discarded lower die rolls to obtain it. Its averages (median = 12, mean = 12.24, mode = 13) are all greater than 10, so it doesn't represent the general population, given the game's stated use of 10 as a typical value for any ability score. Almost half the ability scores produced by this means lie in the range 11 through 14; a score of 17 or better is slightly more likely than a score of 7 or less (in each case, the probability is slightly better than one in eighteen; these are the outer octodeciles). As noted above, this doesn't even match a population of normal people (with typical score ten) in the game world.

So let's look at a distribution that is commonly used in-game to decide the ability scores of the general population: 3d6. This actually has a typical value of 10.5 (that's both its mean and its median; while both 10 and 11 are its mode, so it's also the mid-point between the two modes) rather than ten, but it's used all the same. To get an accurate average of ten one could use 4d4 (10 is mean, mode and median; values range from 4 to 16; its standard deviation is √5 = 2.24; the tails ≤6 and ≥14 each have total weight 15/256, which is just under 1/16); it'd also work to use 1d6+1d12 or 1d8+1d10, but the distributions of these are trapezoidal – flat across the middle, tailing off linearly at either end; one could even use 5d3 (mean, mode and median are 10; range is from 5 to 15, standard deviation is 1.83). Still, I gather 3d6 is usual.

Written by Eddy.