# All Ability Score Arrays



## 77IM

As a public service, I wrote a little program to calculate all the possible ability score arrays using 22-point buy.  You can find it here:  Dungeons & Dragons, 4th Edition: All Ability Score Arrays

But in case my site goes down, here is the full list for posterity.  There are 121 unique ability score combinations.
*
18,14,11,10,10,8
18,14,10,10,10,9
18,13,13,10,10,8
18,13,12,11,10,8
18,13,12,10,10,9
18,13,11,11,11,8
18,13,11,11,10,9
18,13,11,10,10,10
18,12,12,12,10,8
18,12,12,11,11,8
18,12,12,11,10,9
18,12,12,10,10,10
18,12,11,11,11,9
18,12,11,11,10,10
18,11,11,11,11,10
17,16,11,10,10,8
17,16,10,10,10,9
17,15,13,10,10,8
17,15,12,11,10,8
17,15,12,10,10,9
17,15,11,11,11,8
17,15,11,11,10,9
17,15,11,10,10,10
17,14,14,10,10,8
17,14,13,12,10,8
17,14,13,11,11,8
17,14,13,11,10,9
17,14,13,10,10,10
17,14,12,12,11,8
17,14,12,12,10,9
17,14,12,11,11,9
17,14,12,11,10,10
17,14,11,11,11,10
17,13,13,13,11,8
17,13,13,13,10,9
17,13,13,12,12,8
17,13,13,12,11,9
17,13,13,12,10,10
17,13,13,11,11,10
17,13,12,12,12,9
17,13,12,12,11,10
17,13,12,11,11,11
17,12,12,12,12,10
17,12,12,12,11,11
16,16,13,11,10,8
16,16,13,10,10,9
16,16,12,12,10,8
16,16,12,11,11,8
16,16,12,11,10,9
16,16,12,10,10,10
16,16,11,11,11,9
16,16,11,11,10,10
16,15,14,11,10,8
16,15,14,10,10,9
16,15,13,13,10,8
16,15,13,12,11,8
16,15,13,12,10,9
16,15,13,11,11,9
16,15,13,11,10,10
16,15,12,12,12,8
16,15,12,12,11,9
16,15,12,12,10,10
16,15,12,11,11,10
16,15,11,11,11,11
16,14,14,13,10,8
16,14,14,12,11,8
16,14,14,12,10,9
16,14,14,11,11,9
16,14,14,11,10,10
16,14,13,13,12,8
16,14,13,13,11,9
16,14,13,13,10,10
16,14,13,12,12,9
16,14,13,12,11,10
16,14,13,11,11,11
16,14,12,12,12,10
16,14,12,12,11,11
16,13,13,13,13,9
16,13,13,13,12,10
16,13,13,13,11,11
16,13,13,12,12,11
16,13,12,12,12,12
15,15,15,11,10,8
15,15,15,10,10,9
15,15,14,13,10,8
15,15,14,12,11,8
15,15,14,12,10,9
15,15,14,11,11,9
15,15,14,11,10,10
15,15,13,13,12,8
15,15,13,13,11,9
15,15,13,13,10,10
15,15,13,12,12,9
15,15,13,12,11,10
15,15,13,11,11,11
15,15,12,12,12,10
15,15,12,12,11,11
15,14,14,14,10,8
15,14,14,13,12,8
15,14,14,13,11,9
15,14,14,13,10,10
15,14,14,12,12,9
15,14,14,12,11,10
15,14,14,11,11,11
15,14,13,13,13,9
15,14,13,13,12,10
15,14,13,13,11,11
15,14,13,12,12,11
15,14,12,12,12,12
15,13,13,13,13,11
15,13,13,13,12,12
14,14,14,14,12,8
14,14,14,14,11,9
14,14,14,14,10,10
14,14,14,13,13,9
14,14,14,13,12,10
14,14,14,13,11,11
14,14,14,12,12,11
14,14,13,13,13,11
14,14,13,13,12,12
14,13,13,13,13,13
*
One could create a random character that was worth exactly 22 points by selecting an array at random.   Interestingly, 121 is 11 squared, so you could do it by rolling *1d11 x 11 + 1d11 - 11*.  (To make a 1d11, just roll 1d12 and re-roll 12's.)

If anyone notices any errors in the list please post here.  It is very possible that there is a bug in the program.

 --  77IM


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## Jurph

*"Best" Arrays from the bunch*

As a _further_ public service, I threw these arrays into Excel and calculated the stat bonuses (the formula of interest is "=INT(A1/2)+5") that a player can get with the arrays.  There are eight "champion" stat arrays that result in a player getting a total of eight bonus points:

16	14	12	12	12	10
16	13	12	12	12	12
15	14	12	12	12	12
14	14	14	14	12	8
14	14	14	14	10	10
14	14	14	13	12	10
14	14	14	12	12	11
14	14	13	13	12	12

Not surprisingly these are the super-balanced characters who don't spend lots of points at the top of the scale where marginal benefit is less important.  You could use racial stat bonuses to make two of these arrays contain 18s, so a Dragonborn Paladin could have STR and CHA at +3 and +4 (your choice) and take no penalty to other rolls.  I think we all know which races go best with which classes, and which stats are and aren't needed for which classes, but it seems clear that those two arrays up top are pretty nice for all-around gameplay.  If you're not matching your race and class bonuses you may see some weirdness here (e.g. a Dwarven paladin could come out with 14s in key stats and 16s in his racials, making a very tough, very wise paladin who is a bit of a slouch in combat but a hell of a healer).   The balance you get from using one of these arrays combined with the "Aid Another" mechanic means that a character built off one of these eight arrays will be a utility fielder in skill challenges.  If you target your bonus skills carefully you can likely end up very competent in combat as well, since the defenses are all going to be on-par with other level-appropriate creatures.

In the rest of the arrays, there are 30 that give seven bonus points, 45 that give six, 32 that give five, and 6 "stinkers" that only give four (reproduced here to warn off the newbies) :

18	13	11	11	11	8
18	13	11	11	10	9
18	12	11	11	11	9
18	11	11	11	11	10
17	15	11	11	11	8
17	15	11	11	10	9

Even if you use your racials to beef up these ones, you'll find yourself leaning very (very!) heavily on your primary stats and falling flat everywhere else.  The heavy presence of odd numbers (especially 11's!) means that you've wasted points that could be better allocated at a break-point.  Unless you really need to spend points on a 13 to qualify for a feat, you should reconsider your stat distribution if you're using one of these.


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## 77IM

Awesome analysis, Jurph!

You gave me an idea, so I calculated what arrays are available using only even numbers (no points wasted on odd numbers).  It turns out there are only 7 of them:

18,12,12,12,10,8
18,12,12,10,10,10
16,16,12,12,10,8
16,16,12,10,10,10
16,14,12,12,12,10
14,14,14,14,12,8
14,14,14,14,10,10

 -- 77IM


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## Jurph

The "all-even" arrays are much better as a whole than the general population!  There are three "champions" (8 bonus points), and two examples each of 6's and 7's.  There are no 4's or 5's here, so an all-even array is likely to be a good one.  I'm not sure that taking a 7 from the all-even pile is better than taking an 8 that happens to contain an odd number, but it's still interesting to see what a small subset of optimal arrays there are.  The even champions are:

16	14	12	12	12	10
14	14	14	14	12	8
14	14	14	14	10	10

These waste no array purchasing power on odd numbers and give eight bonus points spread around nicely.  A few points in favor of odd numbers:

1) An odd-numbered stat in one of your primary stats would its bonus cross a break-point at 4th level (assuming you put one of your 4th-level points there), but you'd have to wait until 8th level to see the same effect with an all-even array.  However, you can turn that argument on its head and say that you're using clever stat buys to take your bonuses at 0th, 8th, 16th instead of 4th, 12th, and 20th, and I think we all know that an extra point at lvl1 makes a lot of marginal difference.

3) Odd numbers are the threshold for making attribute-based feats possible.  If you only need a stat at 13 to get the feat you want, and you don't use that attribute for anything else, then the points spent from 13 to 14 are wasted.  Flipping through the PHB, 13 and 15 are the two thresholds.  Of course you could take 12 or 14 out of the gate, and bump that attribute at the same time you take your 4th-level feat.

5) Odd numbers can be the sweet spot for things like healing surges and number of hit points.  Once you have 28 hit points, your surge is 7hp and doesn't go up to 8hp until you hit 32 hp total... so a CON score that takes you from 28 hp to 29 hp isn't worth much, but taking you from 27 to 28 hp is a big deal.  For a fighter or paladin (hp = 15 + CON) that means taking a CON score of 13 and spending the residual points elsewhere.  There are only two "champion" arrays with 13s, though, so you might just suck it up and bump to 14, and be content with getting _only_ an extra hit point and an extra healing surge per day.


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## Garthanos

nice number crunching/analysis guys, I didn't think this thread had much hopes of being useful but it really turned around (better understanding may or may not be more fun in play but I like it anyway)


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## Jurph

Just for you, *Garthanos*, here are all of the 7-point arrays:
17	12	12	12	12	10
16	16	12	12	10	8
16	16	12	10	10	10
16	15	12	12	12	8
16	15	12	12	10	10
16	14	14	13	10	8
16	14	14	12	11	8
16	14	14	12	10	9
16	14	14	11	10	10
16	14	13	13	12	8
16	14	13	13	10	10
16	14	13	12	12	9
16	14	13	12	11	10
16	14	12	12	11	11
16	13	13	13	12	10
16	13	13	12	12	11
15	15	12	12	12	10
15	14	14	14	10	8
15	14	14	13	12	8
15	14	14	13	10	10
15	14	14	12	12	9
15	14	14	12	11	10
15	14	13	13	12	10
15	14	13	12	12	11
15	13	13	13	12	12
14	14	14	14	11	9
14	14	14	13	13	9
14	14	14	13	11	11
14	14	13	13	13	11
14	13	13	13	13	13

The PHB notes that if you roll your ability scores and end up with more than +8 or less than +4 in bonuses, the DM should rule that you're too powerful or too weak to play... which makes me think that a DM could, in the interests of fairness (or a challenge) _require_ a party to choose from (say) the 8, 7, or 6 arrays.


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## Garthanos

My group prefers super balanced arrays.


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## Scott_S

I think those "champion" arrays are less useful in practice than they might seem, though.

I mean, look at them:

*16	14	12	12	12	10
16	13	12	12	12	12
15	14	12	12	12	12
14	14	14	14	12	8
14	14	14	14	10	10
14	14	14	13	12	10
14	14	14	12	12	11
14	14	13	13	12	12*

Of these 8 arrays, I'd consider only the first one competitive, because in 4e, having a high bonus in your primary stat is of great importance.  With less than a +3 bonus, you're decidedly weakening yourself.  A +3 bonus is about par, and requires either a 16 or a 14 plus racial bonus.

However, to be really competent, you'll generally want an 18 in that primary stat, which means a 16 in your starting array plus a favorable race.  Only the first two options offer that, and the 16/14 array (offering a secondary +2, three +1s, and a +0) is clearly better than the 16/13 array (offering five +1s), because no reasonable build requires bonuses in all six stats.

(I suppose a bard who really pushed the jack-of-all-trades thing might go for the 16/13 array, but he'd be pretty incompetent with any attack that relied on a +1 stat.)

Now, of the 7-point arrays, we could have:

*16	16	12	12	10	8*

This is one of my favorite arrays.  It provides an 18 primary after racial, a 16 or 18 secondary, and two +1s.  A single -1 is not a problem for most builds.  The 12s will meet many prerequisites by paragon level, and can be bumped a point to meet a 15 prerequisite by epic if necessary (or two points to meet it by paragon).  The tertiary stat will be a little weak without bumping, but many builds don't really have a firm tertiary stat.  In addition, there are enough bonuses to spread around to ensure at least a small boost to all defenses.

*16	14	14	13	10	8*

This is a solid array when a tertiary stat is necessary.  You still get the 18 primary, a 14 or 16 secondary and tertiary, and an additional +1, which ensures decent defenses all around.  The 13 also qualifies for many feats, and will be a 15 by epic without boosting.  The single -1 is again not an issue for most builds.

*16	14	14	12	11	8*

A slight variation of the above.  This one pushes off the "feat qualifier" 13 in exchange for reaching a 12 in the fifth stat faster.  I prefer the one above, but this one is stronger if for some reason you need to place your four top stats into two of the "linked pairs", leaving you with one low defense.  It's also a little better if you need two 13s in otherwise unimportant ability scores in order to qualify for feats.

*16	14	14	11	10	10*

Another variation on 16/14/14, this is generally weaker than the above two, but it's a prime option if you really want to avoid taking a negative.  You get an 18 primary after racial, a 14 or 16 secondary and tertiary, no negatives, and the 11 lets you qualify for feats and (if necessary) get a small defense bonus, with some bumping.

The 18 arrays are also worth considering in some cases:

*18,14,11,10,10,8
18,13,13,10,10,8
18,13,12,11,10,8

*These have very small bonuses overall because of the opportunity cost of getting that 18.  However, they're the only way to start out with a +5 bonus (after modifier).  For builds that rely mainly on a single stat (like many wizard builds), it might be worth the sacrifice.  18/14/11 gives a decent secondary, 18/13/13 and 18/13/12 offer some easier feat qualifications and, possibly, bonuses to each defense.

I don't see any of the champion arrays beating these out in actual play.  16/14/12 might be on-par, but I don't see it as superior.  

Focusing for a couple of strong bonuses is usually the better option in 4e than spreading lots of little ones around.


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## Garthanos

Scott_S said:


> Focusing for a couple of strong bonuses is usually the better option in 4e than spreading lots of little ones around.




And character types that emphasize too many attributes get poked at for being MAD.


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## Rolflyn

Assuming you always add ability increase to your highest stat. At 8th, the 'champions' that produce +9 total mods are:
*15,15,12,12,12,10*
*16,13,12,12,12,12*
*15,14,12,12,12,12*
*15,13,13,13,12,12*​8th level has the same champions as 1st, but they are +10.

11th level there are only two champions at +14:
*15,13,13,13,13,11*
*14,13,13,13,13,13*​In second place at +13, we have:
*16,13,13,13,13,9*
*16,13,13,13,11,11*
*15,15,13,13,11,9*
*15,15,13,11,11,11*
*15,14,13,13,13,9*
*15,14,13,13,11,11*
*15,13,13,13,12,12*
*14,14,13,13,13,11*​At 14th level, the two +15 champions are:

*14,14,13,13,13,11
14,13,13,13,13,13
*​The +14s at 14th are:
*16,14,13,13,11,9
16,14,13,11,11,11
16,13,13,13,13,9
16,13,13,13,11,11
15,14,13,13,13,9
15,14,13,13,11,11
15,13,13,13,13,11
14,14,14,13,13,9
14,14,14,13,11,11
14,14,13,13,12,12
*​The champions of 18th level are the same as 11th.  The champions of 21, 24, and 28 are the same as 1st, 4th and 8th.


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## Llamas Notsheep

Nice analysis   Practically speaking, though, I think things will be a bit different for pretty much any character.

Is there any character out there that's actually going to make use of _all_ attirbutes?  In practice, having a -1 somewhere isn't going to hurt just about any character.  What does it look like if you disregard the -1 for having an 8 or 9 in the array?


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## 77IM

Personally, I hate having a penalty.  Not for any mechanical reasons, but just for role-playing.  My favorite array is 16/14/14/11/10/10, although I like all of the 16/14/14 and 16/16 arrays, for the reasons *Scott_S* describes.

OTOH, I think the 18 arrays and the 14/14/14 arrays can both be viable for some characters.  A warlord, for example, could do OK with 14/14/14 and the right racial mods, and someone like a rogue can really benefit from an 18.

BTW, here are all the arrays without any 8s or 9s in them.  There are 52.  Many are near the "bottom" of the Big List (i.e. starting with a 15 or 14).

*
18,13,11,10,10,10
18,12,12,10,10,10
18,12,11,11,10,10
18,11,11,11,11,10
17,15,11,10,10,10
17,14,13,10,10,10
17,14,12,11,10,10
17,14,11,11,11,10
17,13,13,12,10,10
17,13,13,11,11,10
17,13,12,12,11,10
17,13,12,11,11,11
17,12,12,12,12,10
17,12,12,12,11,11
16,16,12,10,10,10
16,16,11,11,10,10
16,15,13,11,10,10
16,15,12,12,10,10
16,15,12,11,11,10
16,15,11,11,11,11
16,14,14,11,10,10
16,14,13,13,10,10
16,14,13,12,11,10
16,14,13,11,11,11
16,14,12,12,12,10
16,14,12,12,11,11
16,13,13,13,12,10
16,13,13,13,11,11
16,13,13,12,12,11
16,13,12,12,12,12
15,15,14,11,10,10
15,15,13,13,10,10
15,15,13,12,11,10
15,15,13,11,11,11
15,15,12,12,12,10
15,15,12,12,11,11
15,14,14,13,10,10
15,14,14,12,11,10
15,14,14,11,11,11
15,14,13,13,12,10
15,14,13,13,11,11
15,14,13,12,12,11
15,14,12,12,12,12
15,13,13,13,13,11
15,13,13,13,12,12
14,14,14,14,10,10
14,14,14,13,12,10
14,14,14,13,11,11
14,14,14,12,12,11
14,14,13,13,13,11
14,14,13,13,12,12
14,13,13,13,13,13
*​
 -- 77IM


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## Garthanos

77IM said:


> Personally, I hate having a penalty.  Not for any mechanical reasons, but just for role-playing. -- 77IM




heh, I computed my own stats in many game systems back in the day.. til the back injury I had a few years ago.. nothing was below average (Back problems really screw with your stamina - bye bye 12 con hello 8 ). 

No clue how to compute them in 4e but hey ;-)

Why would I want a hero who was gimped? inorder to be super at one thing... monomaniacs are not interesting in real life ;-) however that one thing in D&D comes down to be close to everything.

Makes me inclined to house rule things to make "off" stats more useful.


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## Scott_S

I often like to have one weaker-than-average stat... for roleplaying reasons.  Most heroes have a flaw of some sort, and a wide variety of them are possible to model with a "low" attribute.

Low INT is the only one I try to avoid.  It annoys me a little that in 4e, it's often the easiest stat to "dump"... but I'm a GM who enforces roleplaying the mental stats, and my players know it, so I don't have to worry about it too much.


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## 77IM

Hah, we are opposites, because Int is the one stat that I _would_ set low for role-playing reasons.  I love playing stupid characters.  I'm told I'm quite good at it; I'm not sure whether that is a compliment or not...

 -- 77IM


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## Jurph

Alright, I'm back.  Had to think long and hard about *Scott_S*'s assertion that many small bonuses lack the utility of a large bonus.  I went back to the arrays and calculated the _sum of the squares_ of the bonuses.  These arrays give the top value (21):

18	14	11	10	10	8
18	14	10	10	10	9
16	16	12	12	10	8

and these give a value of 20:

18	12	12	12	10	8
16	16	13	11	10	8
16	16	13	10	10	9
16	16	12	11	11	8
16	16	12	11	10	9

The population of sum-of-squares scores runs from 21 down to 8 and the "champion" (+8) arrays range from 18 down to 12.  

You could also conceivably set the exponents in the simple sum-of-squares formula to be somewhere between 0 and 2 according to your class needs.  That way a score of 8 in your lowest attribute wouldn't matter (the lowest score would be raised to the zeroth power and always contribute 1 to the total) and a high score in your highest would get the full benefit of being squared. MAD classes would need a preference equation that was (approximately) 2,2,1.5,1,1,0 but a class like a wizard could choose 2,1.75,1.5,1.25,1,0 and get a more skewed array.


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## BartD

Great thread 

Jurph, those arrays look like the ones people actually choose. What happens if you don't count (-1)^2=+1 from an 8 or 9 as a good thing?

Also, an odd score is a litle better than the next lower even score since i can be raised or become even in heroic tier. So maybe increase the value of bonuses from odd scores by 1/2 or 1/3 sor omethng like that?

By the way, seeing that...
18 costs 16 points and gives +4 which you count as (+4)^2 = 16,
16 costs 9 points and gives +3 which you count as (+3)^2 = 9,
14 costs 5 points and gives +2 which you count as (+2)^2 = 4,
12 costs 2 points and gives +1 which you count as (+1)^2 = 1 and
10 costs 0 points and gives +0 which you count as (+0)^2 = 0,
it seems that cubed scoring very closely follows the point-buy costs except odd scores are worth less.


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## Jurph

*BartD*, I hadn't noticed that - I've never actually done the point-buys by hand because I've always had a computer doing the substitutions for me.  It makes sense though!

Good catch on that squared negative!  If you want to ignore the lowest score, you can reduce (or eliminate) its contribution to your overall value formula.  For example, my dwarven paladin - who has a serious case of MAD, and so values his second and third stats pretty highly - settled on

FoM = A2 + B1.5 + C1.5 + D1 + E1 + 0

Where A through F are the values of the bonuses given by my highest through lowest ability scores, respectively, and the Figure of Merit is rounded (up or down) by truncating decimals.  I've already prioritized my attributes and added my racial bonuses, so I'm seeing values as high as 22 and as low as 10.

I'm also using a "defense" stat to measure the total points of additional defense I get (values range from 4 to 7) and a "hp" stat so I can see how many hit points I'll have.  Summing all three values -- the merit equation, the defense value added, and my hit points -- gives a range of values from 96 to 79.  Subtract 78 from all of them, and I end up with a merit scale that runs from 1 to 18*... with only _one_ array getting an 18.  The winner (for my case) is:

18 12 12 12 10 8

and the best runners up are:

14	14	14	14	12	8
16	14	14	13	10	8
18	13	12	11	10	8
18	12	12	11	11	8
18	12	12	11	10	9

That first runner-up looks like a stinker but gets two 16s after racials and ends up being mediocre at combat but much more difficult to land a hit on than your average lvl1 -- that's what I get for valuing additional defense and hit points, I guess.  The other arrays also look like they could be useful, depending on your play style.

I think this figure-of-merit calculation would turn out very differently for a fighter, or a mage, or even a character with a different race.  For example, a dragonborn paladin will probably value high all-around defense, but because she has +CHA and +STR, she will probably adjust things accordingly.  A mage might want to spend points for INT, CON, and WIS and either let racials beef up the primary or soak low scores in other areas, so would choose to square the value of the top three bonuses and let the rest drift in the breeze.

* _What are the odds that this figure of merit would happen to end up being the range of character traits in D&D?  Kind of a cool coincidence, but ultimately meaningless._


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