# The Statistical Probabilities of High Ability Scores

The game of AD&D assumes PC race statistics can be found in a bell curve generated by 3D6 for each statistic score. When using more dice or a different method of character generation - other than method #1 - this is not normally to be taken as an assumption the PCs are made of better genetic stock, so to speak, but only to help us gamers generate characters on the higher side of average for play purposes. That is, even though your PCs have high statistic scores that are more probable due to the method of character generation, it should be assumed they, like all of humanity or the other PC races, are a product of the normal statistical variations of the population approximated by the 3D6 rolls. They were just luckier or something, and the PCs are certainly the cream of the crop.

NOTE: NPC population statistic scores are probably not well represented by the 3d6 bell curve. The normal distribution curve, also called a Gaussian distribution curve, far better represents large populations. Thus, some NPC starting statistics (it is assumed starting statistics always represent young adults) might be lower than 3 and/or even higher than 18. For game purposes, however, players are asked to confine their characters to the 3 to 18 range by fiat. That way, while it is possible some NPCs might be higher than 18 or lower than 3, and thus account for such normal possibilities, however improbable, PCs remain in the 3 to 18 range to start so all players may start on an even playing field. Furthermore, GMs may infer racial and/or gender maximum limits on STATS to be for starting characters only, and through magic items, spells, or other extraordinary means, it may be possible for characters to surpass those normal starting limits.

Using 3D6, statistic scores are generated between 3 and 18, inclusive, with an average score of 10.5. Perhaps remarkably, if you multiply this number by 10, it is a close approximation of IQ - the intelligence quotient.

NOTE: The 3d6 curve and the real I.Q. distribution curve are very close to one another near their middles, but the approximation gets worse the further out one goes toward the extremes. More on this, later.

Not to make high IQ sound like the do all and tell all of the statistical picture, but a simple example along those lines will help one begin to grasp where a character's statistics really place them. Whether we are talking about intelligence or some other statistic, the reasoning is similar. To that end, let us consider high school, something most of us have in common.

It will be assumed that the average INT score is 10.5, and the average IQ score is 105, or ten times 10.5. This is very close approximation to reality near the center, but the 3d6 bell-curve and the normal intelligence distribution curve do not fit so well out toward the extremes, yet this analogy is still useful to help one grasp the rarity of higher scores.

One standard deviation is 15 points on the IQ scale. Dice distributions and standard deviations are different for the 3d6 scale, but this is not an article on mathematics, though it heavily uses mathematics.

NOTE: Of course, those well versed in mathematics, or those who know more about the real distribution curve for I.Q., will be able to see just how badly the 3d6 bell curve fits out toward the extremes, and it would be folly to suggest the 3d6 curve and the real I.Q. curve fit well for scores lower than 5 or higher than 16. However, for scores between 5 and 16, inclusive, this is a fair approximation for 99+% of the population. But I'm getting ahead of myself. Let's continue.

Naturally, most people fall within one standard deviation of the norm or average. This means, in this case, 68% of all people have an IQ between 90 and 120 (i.e. 105 +/-15). Average being what it is, this sort of person would gets C's in high school in most subjects.

Moving one more standard deviation out, we will find 95% of all people have an IQ between 75 and 135 (i.e. 105 +/-30). On the higher side, these are the people who got B's and a few A's, while on the lower side we find the ones who got D's and a few F's.

Already we have 95% of all characters falling within the 75 to 135 IQ range. To continue outward for one more standard deviation, we will find 99+% of humanity falls between 60 and 150 (i.e. 105 +/-45). On the lower side, these are the developmentally challenged, as they say, and on the higher side, these are the gifted, the geniuses, the students who seemed to effortlessly pull down A's without even cracking a book - though it probably just seemed that way.

Of course hard work could net someone higher grades than their innate INT would suggest they'd get, or sheer laziness or plain disinterest would have the more gifted pull down less than stellar grades, but for the most part, within the 60 to 150 range, we have seen 99% of all there is to see. To a good approximation, this translates to INT scores between 5 and 16, inclusive. Notice the 10 x factor is not used here. If it were, the I.Q. range of 60 to 150 would be the INT range of 6 to 15, but we use 5 to 16, instead, to make it fit better to the actual percentages. Anyway, we've covered the INT range from 5 to 16, but what about those higher/lower statistics?

NOTE: For scores greater than 16 (or lower than 5), the 3d6 bell curve does NOT fit well to the standard distribution curve for I.Q. (as well as other attribute scores besides intelligence). However, most of this article still applies to about about 99+% of the population. For example, while an 18 INT may occur once in 216 AD&D characters, as they are generated with 3d6, a score of 180 I.Q. points occurs far less frequently than that. Thus, an 18 INT is not the same as a score of 180 I.Q. points. This is because, as mentioned before, the I.Q. curve and the 3d6 curve, though they fit well enough within the 5 to 16 range, are not very good fits toward the extremes. Scores of 3 and 4, or 17 and 18, are rarer in real life than will be generated using 3d6, and, therefore, the 3d6 method isn't really all that great a mathematical model for real world populations, so it would be wrong to think people like Albert Einstein occur roughly once out of every 216 individuals. They are much rarer than that. In other words, while 3d6 works well as an approximation for most of our characters' statistic scores, the 3d6 curve is a rotten approximation to reality when the dice actually generate a 3, 4, 17, or 18.

Getting back to the 5 to 16 range, we find that a 16 gets to be pretty darn impressive in and of itself. Even those students who were always on the honor roles - A or B - may marvel at such intellect. Such a student may even be advanced beyond the grade their age would normally indicate. On the other hand, 5 gets to the point where people cannot realistically take care of themselves.

A very rare 17 is incredible, however. Statistically speaking, even the remarkable person with a 16 INT is frequently taken aback at the fantastic nature of such individuals. Incredible, yes, to say the least, and almost unbelievable at times. On the lower end, 4 is much lower than childlike in simplicity.

18 is pure genius, nigh on godlike in ability compared to most. Though 18 INT and 180 I.Q. are not the same, both are way beyond the vast majority of humanity. Albert Einstein, for example, had an IQ of 180, according to one book I have, and though not unique, such individuals stand out against the backdrop of not only their local group of colleagues, but also pretty much all of humanity. On the opposite end of the spectrum, a score of 3 is probably more like a mindless animal, perhaps operating with only their brain stem functioning. This is as close to brain dead as you can get and still be considered both human and alive.

NOTE: I doubt a character with an 18 statistic score would stand out as much as Einstein stood out, since, using 3d6, 18's occur once in 216 characters for any given statistic, and not once in a blue moon (whatever 5 standard deviations above the norm is, for my tables do not go beyond 4 SDs, but it might be something like once in 20 million, or more). Thus, to account for this, it is assumed starting scores above 18 are reserved for NPCs, or for those who somehow come into such lofty scores later in life, through further development, training, or magic.

So what does all this mean for the game of AD&D? I think it means, or should mean, with a greater understanding of the normal statistical distribution curve, you may gain a greater appreciation for what it is to have a statistic outside the 9 to 12 range.

On my world, I don't even let adventurers of any class START with statistics at 7 or below. They would be too developmentally challenged, and no one would bother to train them. Even if they could learn, and if they were trained, they'd almost certainly be killed quickly without someone else looking out for them nearly every step of the way. But that is a PC consideration. An NPC may be taken in hand and nurtured along the right lines, but they will never have the right stuff, and never be able to survive on their own against the challenges that one finds in the field. Therefore, I will speak no more of statistics on the lower end of the scale.

Now that we begin to appreciate how impressive a 13 really is, we may be less inclined to expect or even demand an 18 - or even a 17. And that's for our PC's primary requisite, let alone the others. Besides, once you start to play PCs with statistics at 17 or higher, you are getting into the realm of the unknown for most of us. Playing a wizard with an 18 INT, or a priest with an 18 WIS, or a paladin with an 18 CHR, for example, may be difficult even for a player whose own intelligence, wisdom, or charisma is pretty high - 120 to 140 or even to 150, for instance.

With most of the other statistics, dice are required and expected to tell the tale. That is, a character with an 18 STR needn't act strong since the dice and STR tables will reflect their Herculean strength in the game, and an 18 DEX or 18 CON is similarly handled by dice; they needn't act 'quick' or act 'healthy.' 18's in INT, WIS, or CHR, on the other hand, may actually require you to play beyond your own abilities - and in many instances, play beyond your own understanding or persuasion. In short, unless your own statistics fall close to your character's high INT, WIS, or CHR scores, you will not be playing that character as realistically as you may think, thus necessitating many more STAT rolls or GM hints to aid you in your play.

For example, if the GM made a simple logic puzzle or some mathematical trick, your wizard should be able to easily handle it. But what if the player of that wizard has no gift for logic? Should the GM just give them the answer when asked?

The argument that their character could so EASILY solve THAT particular puzzle such that it shouldn't even require a roll is frequently made. And I mean, unless the GM also has an INT on par with the wizard - perhaps genius or above - any puzzle the GM made is probably simple enough so the wizard, ostensibly of higher intelligence, could automatically solve it, and therefore no roll should be required.

Be that as it may, the GM may at least feel compelled to give the player of the wizard or other highly intelligent character an INT roll to solve any problems of intelligence - the player's own real INT be damned. Similarly, the GM may feel compelled to give the players of priests or characters high in wisdom a WIS roll to solve social or common sense problems, and the players of characters with high CHR a roll to persuade both PCs and NPCs alike to follow them blindly, if need be, no matter how unimpressively they may play their characters. This sort of takes the fun out of ROLE playing - though it's fine for ROLL playing or computer games, where there is little, if any roleplaying at all. So where does one draw the line?

It will ultimately be up to your GM to decide how to handle these problems. Personally, I feel too many INT, WIS, or CHR rolls spoils the point of roleplaying, so if my players play beyond their means, tough. My point here is to just state flat out that if your characters have more reasonable and/or 'probable' statistics to begin with - in the 6 to 15 range - this will not be as big as problem as it may be otherwise. So if GMs wish to avoid many of the common pitfalls of higher statistics, please simply keep the runaway stat climbing to a minimum for your game.

Remember, to enjoy a social game like AD&D, it isn't necessary to be a walking god or have statistics that are far beyond the norm; it is usually more than enough for your character to have statistics similar to the other player characters. With all of you in the same boat, you can enjoy the game without feeling shortchanged.

One way I find that may help is to allow a PC with high INT, WIS, or CHR, to draw upon the other players, even if the PCs of these other players are not in a position to help. For example, if a wizard with high INT is alone in a locked room and needs to solve a logic puzzle to escape, the GM should allow the other players to help him even though their characters are not in the room with the wizard.

NOTE: That wizard's player should still ASK for help first, since they may wish to do it on their own. Blurting out the answer without that player asking for help may anger them, particularly as your character is NOT there to help. Thus, the GM should only allow other players to augment a character's high stat when the player of that character asks for help, and not before.

This tactic does a couple of nice things. First and foremost, it augments the wizard's intelligence and allows the character to act more realistically. But as a bonus, it involves all the players and lets them all play and contribute instead of sitting on their hands or biting their nails while feeling helpless or frustrated that they can't help or participate.

But mostly the GM needn't feel like he or she has to give that player any help simply because that PC's stat is much higher than the player's own. I hate it when GMs have to solve their own puzzles. Far better to have the players do it for themselves. Otherwise, there is no point in actually making a puzzle, is there? Just state there is a puzzle there of some type, and declare the DC of the puzzle is such and such and let them roll vs. the appropriate stat. But then, that is roll playing, not roleplaying.

Similarly, high WIS or high CHR may be augmented by allowing the other players to coach another player, upon their request for help, about what to say, what to ask, how to say it, or other things like that. Ultimately, if the GM makes problems or puzzles, he or she wants the players to solve them and shouldn't have to do it for them, and shouldn't allow simple dice rolls to solve it, either. This technique will help.

One might also bear in mind the following. If the GM is making a puzzle some NPC made - and this NPC has an INT higher than the GM's INT - in truth, the GM's puzzle only represents the real puzzle found in the game world. You might call it a poor translation, in fact. Why is this important? Because the argument 'My highly intelligent wizard could easily solve THAT puzzle,' no longer holds.

For example, the GM - INT 14 - makes a real puzzle to put in his game, but that puzzle was supposed to have been made by a sage - INT 18. So the GM's puzzle is, at best, a poor imitation of the sage's real puzzle. Now along comes the player of a wizard, where the player has an INT 13 and his wizard PC has an INT 16.

The player says, "My 16 INT wizard could easily solve THAT puzzle," and he'd be right since that puzzle is not the real 18 INT puzzle in the game world, but only the 14 INT made by the GM. Do not fall for this player's line of reasoning. The real puzzle is much tougher, and worthy, therefore, of an 16 INT wizard to the SAME degree as the GM's 14 INT puzzle is worthy of the player's own 13 INT - or close enough, as a first approximation

Now I'll finish this article with some raw numbers about characters with high statistics. These probabilities are based on 3D6 and concern starting statistics for player characters only. Though they most often will, NPCs need not necessarily conform to these standards, as freaks of nature can and do occur, but to be fair, PCs must conform to the rules of character generation. Remember, after you begin play, magic and happenstance can alter things beyond normal starting consideration. Thus, the actual statistics for the 3D6 dice are being used, and these are given in the table below.

For any given statistic, only one in 216 characters should even have an 18 in that statistic. For example, only one in 216 characters will have an 18 INT. Yet, you will discover, it is far easier to roll this character up. Why? For the simple reason players are allowed to roll up 6 statistics and place them in any order they wish. Thus, since one in about 37 characters will have at least one 18, simply shifting it to the INT slot makes a character with 18 INT, and one need only roll up around 37 characters to get an 18 INT wizard. This, unfortunately, may lead one to think an 18 INT character is one in 37, while in truth, 1 in 216 better reflects what's actually happening in the population.

For any given two statistics, such as STR and CON, for example - (great stats for fighters) - having two 18's in those particular positions is about one in 46,656. However, when allowed to float one's stats around freely, you will find it only takes about 3,100 or so characters to find one with two 18's, which a player may freely place in those positions. So are high STR, high CON fighters one in 3,100 or so, or one in 46,600 or so? The latter better reflects the truth, and demonstrates the rarity of a particular combination, no matter how deceptively common certain rolling techniques may make such characters appear.

This makes impressive characters of a particular class, for example, pretty rare, in actual fact - far rarer than the rolling methods might otherwise lead one to believe. So while it may only take relatively few actual attempts to roll up a particular character when using this stat shifting method, the truth is such characters are much rarer than that.

For example, 1st edition monks have the following minimum requirements: STR 15, WIS 15, DEX 15, and CON 11. This exact combination, or better, comes up about once in 2,500 tries. However, when allowed to shift stats around, it might occur as frequently as once in 78 tries or so. Are characters in the general population that qualify as monks, therefore, one in 78, or one in 2,500? The lattermost, obviously.

Rangers, too, are hard to qualify for. 2nd edition rangers have minimum requirements of STR 13, DEX 13, CON 14, and WIS 14. This exact combination, or better, comes up about once in 568 tries, or there abouts. However, when allowed to shift stats around, it might occur as frequently as once in 27 tries or so. Are characters in the general population that qualify as rangers, therefore, one in 27, or one in 568? Again, the lattermost, obviously.

NOTE: In actual fact, such characters may be even rarer than this, as I did not take into account disqualifying factors. For example, though one in 568 characters may have the high, minimum stats for a ranger in those four key areas, some few of those might also have, for example, accompanying stats of very low INT or very low CHR (maybe even at a developmentally challenged capacity, such as an INT of 3 to 7). Such disqualifiers, however, are probably pretty rare, and I think we may safely ignore them without worrying too much about being that far off our one in 568 estimate.

Taking at least a minute to roll a character, and working about 12 hours a day, it could take well over two years to get a single lofty character with three or more 18's in the right spots. To get lesser, but still qualifying characters for some of the harder classes, however, still could take days, if one actually rolled them up. Of course, using a computer to generate them by the bushel would only take scant seconds. I just rolled one up with four 18's, for example, in under 15 minutes, and even that's almost a fluke amongst the 100 MILLION characters required to do it. My point is, if one is going to roll up that many characters, one may as well simply take whatever they want and skip the bother of actually rolling them up. There is no particular virtue in actually having a computer roll it up.

NOTE: The four 18's character mentioned above was the type where order didn't matter and one could shift the stats into any four desired positions. However, just for your own edification, if four particular stats were desired, that character would be one in over 2.1 BILLION characters. Perhaps 2 or 3 such individuals actually exist on Earth, with a population of over 6 billion people, but I digress. Just keep in mind, when the exact stats are desired, the rarity sky rockets far beyond the relative ease it may take even for a super computer to generate them.

In the past, even I had the unfortunate tendency to allow my players to have characters with one 18, one 17, one 16, with three average statistics in the other areas, though I'm trying to cut back on this habit. Now I prefer letting them have a 17 or 16, a 16 or 15, and a 15 or 14, plus three remaining stats such as 12, 12, and 11 or 10. These are still rather impressive characters, far above the norm, you understand.

However, let's go back and assume an 18, a 17, and a 16 comprise a character's best stats. Such a character would only be one in 558 or so, when allowed to place the stats wherever one wished, yet even here, if a particular order were required, as often is the case when you have your heart set on playing a particular character type, when rolled in a particular order, and to get a better feel for the rarity of such a character within the general population, it would be far more difficult to achieve a particular desired class or class combination than 558 tries. In this case, to have an 18 in a particular stat, and a 17 in another particular slot, and a 16 in yet another particular third spot, this only occurs once in over a quarter of a million characters. So, for the last example, a rogue/wizard with an 18 INT, 17 DEX, and 16 CON, or better in these areas, occurs only once in 260,000 tries or so. Of course you need only roll around 558 characters, if allowed to arrange the stats as you wish, but never forget, in regards to the general world population, that character is roughly 4 in a million, and not 2 in a thousand.

Anyway, in a medieval setting where populations may be a bit low - Alodar, for example, has a population of 750,000 within its walls - there should only be one or two such characters who arise every generation or so, and these are your typical heroes, kings, or characters of fiction. True, perhaps 1 to 2 thousand characters exist with an 18, a 17, and a 16 within Alodar's walls, but they could easily be arranged in some unfortunate manner, or perhaps combined with an equally improbable, yet remarkably bad stat, such as a CON of 5, a WIS of 6, or an INT of 7. Such combinations would almost certainly disqualify a character as a worthy PC, or even a formidable NPC. All of this makes impressive and worthy characters far rarer than the relative ease rolling them up and shifting around the numbers might suggest.

Unfortunately, many AD&D supplements have felt the need to give such legendary characters two or more 18's, 19's, or possibly 20's or more! I say this is unfortunate since it lends credence to such improbabilities as being almost necessary for one to be heroic. The truth is, such legendary characters would have been just as impressive and just as heroic with one 18, a 16 or two, and a 15, along with pretty average stats in the remaining slots, particularly when compared to the masses that surrounded them or populated the land - i.e. normal folk with normal stats who wrote about them and were the ones being impressed. But you can read more about my take on that subject by following the link below:

The Over Estimation Of Our Literary Heroes (Why Are They So Powerful Anyway?)

Of course, if you allow for godlike genetic bloodlines here and there, and other racial bloodlines in the mix, and if you allow for an influx of off-worlders coming to your world, there could be more of these than one might normally expect.

Well, this is fantasy. Just don't think this means one should ignore normal probabilities and take or make whatever you feel the GM will let you get away with. That would be unrealistic. You should police yourself, even if your GM is reluctant to do it. If nothing else, the character will more easily be able to migrate to another GM's world later without causing your GM to initially wretch. But more importantly, the trend away from mini-maxing makes the game more realistic and believable, and, as it turns out when you think about it, mini-maxing often doesn't help you achieve greater things, anyway. To see why that may be, you can follow the link below:

The Problems Of Mini-Maxing (Why Mini-Maxing Is At Odds With Good Roleplaying.)

NOTE: Remember, this analysis is based on mortals, human or typical PC racial genetic stock, and STARTING statistics. Thus, a king or a hero may actually have some pretty impressive stats by the time they achieve such greatness, but this is usually the product of special training, blessings, or even magic. Also remember, if your PC has stat adjustments - like +1 here or -1 there - this analysis is only meant to apply to what they actually rolled and not what they ended up with. So there will be slight differences here and there.

Well, at least we may take solace in the fact that our PC groups probably should not only be the cream of the normal crop of characters world wide, but perhaps the cream of the adventurer crop as well - PCs just naturally tend to be better than most NPCs. After all, isn't the GM likely to put some of the world's greatest challenges before your group, and shouldn't they be able to cope with them? This is not to say some pretty impressive NPCs aren't out there, but they should be rather rare, statistic wise.

And, as always, good fortune and hard work and more time may produce characters beyond where you'd normally expect by just looking at their stats, so the occasional overly impressive NPC is fine; just don't over do it when NPCs whit more average looking statistics will more than do the job. For example, a high level wizard needn't have an INT of 18 or more to kick ass, but may just as well have an INT of 14, 15, or 16, and be the product of ten to twenty years more experience, and be older than most PCs. There is little need to make them a walking godling amongst men to make them formidable.

NOTE TO GMs: It may be hard to prevent your players from having highly improbable statistics if they tend to bring the character with them to the game - into your world from a previous game - but if you do have the luxury of starting out at square one - that being actual character generation - then please try to aim for statistics at or below 16 for their primary requisites. In my IRC game, the reason I was on the higher side of probable is because many of my players came in with those characters, so to remain on par with the other PCs, even homegrown ones had to be that high to keep up.

As GMs, we all know if the party is at an elevated level, we just have to throw elevated encounters at them. Before you know it, we're in the midst of the ridiculously improbable, even for fantasy. After all, probabilities should apply to your monsters, too. It isn't necessary to play in the realm of the improbable in order to have fun, and if you take this to heart, perhaps one day we will be pleasantly surprised to find migrating characters whose statistics are actually more believable.

I say, if a player shows up with a realistic character, they should be rewarded throughout the run of the campaign, and if they come in with the highly improbable, they have already taken their reward and should fight for everything else they may ever get while on your world. They wanted those stats, and it will be up to them to live up to their promise and their character's potential. If they can do this, their rewards will come, and if they can't. . . well, let's just say they would have done better with a more realistic character.

Finally, below please find a simple table of 3D6. The table indicates a statistic score, the probability of rolling that statistic on 3D6, and the percentage of characters who should have a statistic equal to or below that number, N.

For example, rolling a 15 for a particular statistic will happen 4.63% of the time, and 95.37% of all characters (with starting statistics) should have a score of 15 or lower for that statistic.

 STATISTIC Probability % The % <= N STATISTIC Probability % The % <= N 18 0.4630% 100.00% 10 12.500% 50.00% 17 1.3889% 99.54% 9 11.5741% 37.5% 16 2.7778% 98.15% 8 9.7222% 25.93% 15 4.6296% 95.37% 7 6.9444% 16.2% 14 6.9444% 90.74% 6 4.6296% 9.26% 13 9.7222% 83.80% 5 2.7778% 4.63% 12 11.5741% 74.07% 4 1.3889% 1.85% 11 12.500% 62.50% 3 0.4630% 0.46%