[Drum roll] Statistics

Bell curves

As we all learnt in the DMG (page 10) 3d6 creates a curve and calls it a bell curve. But this curve is not normally distributed and is therefore not a true bell curve as often suggested or used inappropriately.

Look at the comparison of the two different curves below:

3d6 probability in red and a normal distribution bell curve in green (apologies as hard to see somewhat)

http://www.netmba.com/statistics/distribution/normal/ gives a good summary of Bell Curve Characteristics and I’ll Compare them to the 3d6 curve above:

Symmetric [Yes]

Unimodal [Yes]

Extends to +/- infinity [NO]

Area under the curve = 1 [Yes, since all values of 3d6 are under the curve, unless one allows race and age adjustments beyond 3-18, when the answer would be NO]

So close, but no banana.

The Empirical Rule for a normal distribution bell curve is:

68% of the data will fall within 1 standard deviation of the mean

95% of the data will fall within 2 standard deviations of the mean

Almost all (99.7%) of the data will fall within 3 standard deviations of the mean

Torben, at the Troll Dice Roller, has an excellent little free on-line program to calculate probabilites. (He’s very helpful by the way and sent me the code for 3d6 reroll 1’s when I emailed him, which is: “sum 3#(repeat x:=d6 until x>1)”.)

Anyway the probability for

**and capturing his screen shot is:***3d6*The spread or standard deviation (SD) for 3d6 is 2.958 (or 3 amongst friends) from a mean of 10.5.

1 SD around the mean is 7.5 – 13.5, which has only 60.6% of values compared to normal distribution 68%

2SD 4.5-16.5, 97.7% vs 95%

3SD 1.5 – 19.5

That is, it looks exactly like the red curve at the top of this post but shorter and plumper, with less tail than a normal distribution.

Since as stated above, almost all values will lie within 3 standard deviations of the mean and since one can’t generate on a 3d6 ability scores of 1,2 and 19, 20 the point that a 3d6 curve is not normally distributed becomes obvious.

So what? Daniel Collin’s essay on this whole IQ, Intelligence score and bell curve topic is well worth reading. One argument he makes is why should our fantasy world follow the same rules for normal distribution as our real world. Very true and difficult to argue that one.

He also says the following and I’ll quote a large chunk but please hit the link to read the rest:

“One of the most popular alternate theories by D&D gamers who are unaware of this history is to theorize that they should "compare the bell curves". That is, they contend that one should calculate a relation by considering what percentage of real-world people have a certain IQ range (constructed specifically with a 100 mean, and standard deviation of 16), and map that to a range of equal percentage likelihood when rolling 3 six-sided dice (range of 3-18, with a 10.5 mean, and standard deviation of 2.95). The end result is a formula such as Int = (IQ – 100) / 16 * 2.95 + 10.5.

A relation like this has the effect of scaling the extremes of IQ scores further out on the Int scale. This presents several practical problems: (1) animal-level intelligence would correspond to Int 1 = IQ 48 or Int 2 = IQ 53, which would be high enough to learn language; (2) the minimum for humans, Int 3 = IQ 60, is sufficiently high as to entirely miss several categories of real-world intelligence deficiencies (see below); and (3) the maximum for humans, Int 18 = IQ 141, is actually far below the results for some real-world people on standardized IQ tests.

The best example of this last problem is the Guinness record-holder for Highest IQ, Marilyn vos Savant, who reportedly has an IQ score of 228 (subject to some debate; see links at end). Under the "compare the bell curves" theory, this would translate to Int 34, which is wildly beyond the range possible in D&D by rolling 3d6 (in fact, beyond the range of most gods in D&D). Even if we are skeptical of this IQ score, considering the previous record holder’s IQ of 196 results in Int 28, again far beyond the 3-18 result achievable in D&D. In contrast, the simpler linear relation properly brings these scores into the more reasonable range of 19 and 22, which is in fact naturally achievable in D&D via several methods. (E.g., a roll of 18 plus a few age or level-based ability bonuses, reasonable point buy, etc. The "compare the bell curves" method is not remotely correctable even by maximizing such increases.)

One amusing "advantage" of the scaled curve system is that it strokes D&D players’ egos by making them look extremely smart in game terms. I’ve seen multiple online discussions of this topic in which everyone participating gleefully points out that their IQ scores translate to a D&D Intelligence of 18 or more under this model, and feel that that’s entirely reasonable.”

*Thanks Daniel, back to me:*

So bell curving D&D ability scores is poor stats. If we are going back to intelligence it tells us two things, we don’t all have IQ of 18. Embarrassingly, I once used my state-wide education academic record that said I graduated in the top 1% to conclude, ah-hem, I had an intelligence of 17 - 18, so I was very guilty of the poor thinking Daniel is criticising. I would estimate 11-12 now (IQ 110-120), and I’m considered clever. What this should tell us about low Intelligence scores is that an Intelligence score of 3 is really, really bad. If strength, wisdom, dexterity, constitution and charisma in any way mirror intelligence, then the same should apply. More on other ability scores in the next post.

But first a way out:

While not completely necessary for my argument this is a useful link on IQ and standard deviation. I could, at a pinch support the fudge (very bottom answer – ‘compressed IQ to 10 points per INT past 1 std dev’), that redefines IQ to not be as bad or good at the extremes – therefore a playable ability in the 3-18 range.

However this would not support Argument 1 (first post) where I showed the rules clearly relate Intelligence score of 3 to IQ 30, which is unplayable. So either way one is house ruling. You can’t have a playable character with IQ 30 and must reject it (breaking the rules that say 3-18 is playable), or one is forced to accept a new distribution of IQ that was never suggested or supported in the rules. You choose?

Reroll, readd, redo - An ability score of 3 is unplayable

PS I need to thank Spawn of Endra. In my

**first posting on this topic**I said, "Intelligence is the only ability score that is related in the rules to a real life comparator." Spawn of Endra pointed out something I had overlooked in the Players Handbook. Strength of 3 = the ability to bench press 30 pounds, 18 strength, 180 pounds. This needs more to be said, and I hope to get to other ability scores soon, as promised.
While a bit math-heavy for me in parts, I really appreciate this analysis, and very much look forward to your future posts on other attributes!

ReplyDeleteAhoy Mr. Priest,

ReplyDelete{I have not fully digested your statistical arguments yet, and will likely comment on them soon, but if you are considering your Strength post, here are some new thoughts.}

Thanks for the acknowledgment, most of this AD&D technical savvy is due to me reacquiring my old copies of the books, not innate Old-Schoolness. But in a further screw-twisting I'll clarify that the Strength entry in PHB p.9 specifically refers to lifting a "weight above his or her head in a military press", rather than a bench press. The dreaded 'military press' is a seated vertical press, I believe, whereas the standing press is a bipedal affair.

In general the standing or military press always uses a lower weight than a bench press. How STR would relate to encumbrance using these weight values is another intriguing question.

More later. Yours is a rich post.

@ Spawn of Endra

ReplyDeleteYou're totally right about the military press - this time it was a simple mistake as I meant to say military press not bench press, as per DMG, unlike the hole in my rule reading you pointed out last time.

Thanks to you and Carter Soles for your loyal commenting to my ramblings.