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.

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.

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 …

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.