# Is it better to take the array and be Joe Average, or to roll for the odds of getting on average better scores?

I am new to D&D. I was looking at character creation for D&D 5th edition. There were a few ways one could generate ability scores. I assumed the optional method of taking the numbers 15,14,13,12,10,8 would be at least as good as the default chance method (roll 4d6, drop lowest die), and more likely, just a bit better than chance.

However, with the above method, the summed ability scores is 72, which is just a bit shy of the summed average one would obtain by rolling dice: the average ability score generated by dice should be 12.2446, which means the sum of the average ability scores is 73.4676.

What the 15,14,13,12,10,8 method accomplishes is to give some moderately high scores, but no exceptional ones, without giving any terrible scores (dice rolling typically gives at least one score of seven or less). To my mind that suggests that the rationale is that many players may find that "joe just-below-average" across the board is better than Achilles, who is amazing in some ways, but has that crazy heel weakness. He is also maybe better than "Joe Exactly-average" who has no high scores and no low ones?

Are those rationales good, i.e. is it actually better to have Joe Just-below-average-with-some-bright-spots than Joe Completely-average or Achilles?

Correction: The odds of rolling all ability scores at 8 or above are 70%, so I misspoke when I said usually one will roll one score below an 8. In fact, usually one does not, but not in a strong sense. It is no more unlikely to get a score below 8 than two coin flips coming up tails. It happens.

I provide a quick chart at the end, which makes the statistics easy to generate. As I generated the numbers quickly, I confess the possibility of error. For the 1296 possible rolls of four dice here are the number of ways you can obtain each value as the sum of the best three.

Sum of best three | Number of possible rolls that give that sum of best 3
--------------------------------------------
3                     1
4                     4
5                    10
6                    21
7                    38
8                    62
9                    91
10                  122
11                  148
12                  167
13                  172
14                  160
15                  131
16                   94
17                   54
18                   21

• Whippersnapper! The array doesn't yield Joe Average, it yields Joe Hero! 4d6-drop-lowest is already a huge advantage over real random stat generation - 3d6 in order. Kids these days! ;) – gomad Sep 23 '14 at 11:49

## It is better to take the array.

There are several reasons for having all players start with the same points then choose their stats:

1. Consistency - You'll never have to worry about a player who rolls poorly being stuck with bad scores across the board.

2. Fairness - You won't end up with any players rolling really well and having multiple high scores. Characters with those ability scores can often overshadow others.

3. No Cheating - You won't have to worry about players fudging their dice rolls.

4. No Upset Feelings - You won't have to deal with any player jealousy's over another player having better ability scores.

• 5. Players won't "suicide" to get a character with better stats. – Tijnkwan Mar 14 '15 at 23:38
• @Tijnkwan I fondly remember a campaign in which, during the first game, I had a player jumping several times down a cliff (or charge an orc, or burn himself to ashes) and create new characters until he was satisfied with the roll he got. It was a great laugh. Thing is, for me to allow him to re-join the game and kill himself again he had to create the FULL character, including feats and calculating all stats (he played about 15 minutes of a 5 hours game). It became a great gag. (And later the dead returned as undead villains) – xDaizu Jun 20 '17 at 9:53
• The only problem I have with this answer is that it appears to be from the PoV of the DM, while the question was asked from the PoV of the player generating a character. (All 4 points are certainly considerations that may arise at some tables ... but won't at others). As a "general answer" to a table full of new players, pretty solid advice. – KorvinStarmast Mar 8 '18 at 13:12

This blog post has some AnyDice scripts for the 4d6 drop lowest stat generation. Looking at the results, the average rolled array should be 16, 14, 13, 12, 10, 9.

That's pretty close to the 15, 14, 13, 12, 10, 8 standard array, although slightly better. So really they're likely to give you about the same results overall. Roll if you want the chance for higher at the risk of lower, but odds are it'll be close to the same.

As for the chances of rolling higher or lower, I think it's best visualized by this graph. Each line shows the chance of rolling that or higher for that score. With Ability 1 being your highest roll and Ability 6 your lowest. There's only about a 15% chance of rolling an ability below 8, with about a 56% chance of getting at least a 16. Getting an 18 about 9% of the time. I'd say rolling is probably in your best interests, you probably won't roll drastically lower and just might roll higher. The standard array is still a decent 'safe' option though, if you don't want any risk at all.

The Script in question for posterity

ABILITIES: 6 d [highest 3 of 4d6]
loop P over {1..6} {
output P @ ABILITIES named "Ability [P]"
}


It depends.

As you've pointed out, with the rolled average you do end up with (on average), slightly higher scores. But you also distribute them more towards the center.

As you can see, there is a legitimate possibility that you'll get all of your scores at or below 13 when you roll. And even if you get one high score, you've still got the possibility of rolling abysmally for the rest of your scores.

Others have mentioned another potential issue as well. If one guy at your table rolls well, and another rolls very poorly, there could potentially be enough of a skill gap at the table to be a burden for the players and provide an unfun experience to the group. This is a big part of the reason why this randomness was eliminated in 4e and why a lot of tables prefer the array.

So do you want to trade guaranteed competence in 2 skills (what you put your 14 and 15 in and hopefully your race bumps them both to 16) for the remote chance of a higher than average starting score (17+ post racial shows up pretty rarely, but not crazy rarely)?

That's not a trade I'm willing to make, but it's one a lot of people are interested in. It's also probably worth mentioning that other than saves and skills (not that these areas should be overlooked, they are important), you can be a competent adventurer for most classes with a single strong score, so rolling, provided you get at least 1 14 or better score, is not the end of the world that it might be in other editions.

The other trade off is that you're trading an absolute minimum score of 8, for the remote possibility of rolling a 3 on an ability score (granted that's a 1 in thousand or so chance, but it's still on the table) and the possibility of rolling at least 1 score below an 8.

So while the rolled array does provide a slightly higher average score distribution, it's centered, and carries a good bit of risk. You risk more mediocrity with rolled scores than you do with taking the array.

### For public/organized play, rolling is off the table

Important note: If you're playing organized play, you are required to use the standard ability array or point buy. Rolling is not allowed.

Math aside, I think there's a perception problem here. All adventurers are way above average (as are some significant NPCs). Take a look at the Commoner on page 345 of the Monster Manual — a 10 for every ability. That's the baseline. In many game worlds, that's going to be the vast bulk of humanity, with some people below the curve and a few exceptional people — heroes and villains — above.

Achilles very well might have had a (starting) strength of 15, plus a bunch of levels in fighter and, you know, that heel thing.

To go back to the math, the main risk with rolling isn't that the mean is better; it's that you're not making enough rolls for everything to really average out, and the curve isn't all lumped in the middle or high end, leaving you with a pretty good chance of 6 terrible abilities.

But that's not all. The limited number of rolls comes in another way: in comparison to the other people at the table. Because this is a group game, if you roll super-well and a couple of people don't, that's basically like choosing at random to make some people have a penalty to all of their rolls for the whole campaign. To me, that's less fun — at least for a long campaign where characters last a long time.

I know some people really like it even then, but to me, rolling is best when you are playing a game with new characters frequently: either because you're playing many one-shot short adventures (in which case it can be fun to roll first and then figure out what character concept those numbers represent), or because you have a high-mortality game where it's common for characters to die. (In that case, the rolling method tends to higher numbers by natural selection — the characters with weaker stats die off sooner and, then, hey, maybe the next one has an 18 con, and that guy doesn't die so easily.)

But wait, that's not all! In this edition, ability scores are improved as you level and capped at 20. This means that a theoretical high-roller will be ahead of the pack at low levels, but after a handful of levels someone who took the array could easily catch up on the primary stat. And, you have to increase by 2 in order to be meaningful, which means that the mechanical gap is small. The die roller will be left filling in non-primary stats (generally, nice but not a big deal) or taking feats (which tend to be awesome, but also situational rather than applying to most checks). And, in 5e, you have saves attached to all of your abilities, which makes that Achilles' heal more of a risk — you can't just put that 4 in Charisma and ignore it.

So, anyway, which is better depends on what you want (and that's why both ways are options). But overall, I think 5e has done a good job of making it not a huge deal.

I prefer rolling stats instead of using arrays for two reasons.

1. Stat arrays make duplicating characters almost insanely easy. Mr. Joe Schmoe wizard suddenly dies in a dungeon. Now you have Joe Schmoe II heading into the same dungeon his player's previous character died in. And he dies again and he goes in again as mark III, IV, V, VI, VII, and VIII. Dies all eight times and doesn't change a single thing on his character sheet besides adding another tally for number of times his guy has died. But he doesn't have to change anything because he uses the same pre-approved array the group uses and by RAW this is legal because WBL lets him "buy" the items he already has when making a replacement character.

2. Any half decent group will let you re-roll anything below a 7. And most groups will erase a character with three 18s because of group balance. And besides, what's more fun: Gambling with the dice over how powerful your character will be or using a below-average array that makes everyone feel like a commoner with slightly higher stats?

It's up to preference, but I like rolling because it tends to add flavor to a campaign instead of the cardboard feeling of using pre-made characters and stat arrays.

• If this had been posted before an answer was accepted, i may have selected it as the best answer. Point number 1 is excellent. – John Robertson Dec 30 '14 at 21:40
• Forcing players to roll stats seems like an inappropriate solution to the problem of players creating carbon copy characters, and is probably better solved by talking to the player in question. It's not like rolling stats is likely to change much when you're stripping out terrible and excellent rolls for "group balance" purposes. – dhmstark Apr 9 '15 at 23:29
• @dhmstark If you read the 5e rule book, the roll 4d6 drop 1 is the default, with array and point buy as options. Your "forcing players to roll" is at odds with how the game was written. – KorvinStarmast Mar 8 '18 at 13:14
• Rolling for stats may be the "default", but it is also a legacy method that causes problems detailed in other answers to this question and any DM worth their salt will at least allow players to use a different method. Indeed - in AL, it is required to use a different one. This is an irrelevant point to argue about, however. This answer is positing problems that rolling might (and I suggest won't) solve. – dhmstark Mar 31 '18 at 12:39

The answer is truly... it depends. Rolling is, of course, more variable, but there is more to it than just risk vs balance.

On average it works out to be about the same or maybe only slightly better to roll. The thing is, do you want the chance to do better than the standard array? As a player, my answer is always yes, but I don't sweat it when the table decides to use the standard array. In the end you still get to play the game. As a DM, it's a bit trickier to answer. In order to make sure that everyone at the table gets to contribute to the game, it might seem better to go with the standard array for everyone. As I already noted, I don't sweat it when I'm playing and DMs/Tables choose this. That said, in my experience the lack of balance that others complain about hardly ever surfaces in home games with mature players. Things might play a bit differently at a public/convention environment.

I prefer rolling, even if I end up with sub-par stats because rolling forces me to make decisions about my character that I wouldn't normally make when using point buy or the array. People talk about having stats that are even numbers so that they can capitalize on the appropriate bonuses, but rolling takes that decision point away while giving you others that make a character more fully realized. For example, I might have a Fighter where Intelligence is a "dump stat", but with rolling I might end up with an 11 or more there. I could certainly have done that with the array, but it's not what I would tend to do. I would tend to point the points where it makes sense to make the most effective fighter. Now let's say that the 11 Int triggers some desire to do something with Int, I might at some point later even increase that Int to 12 with an Ability bump. I might not have ever considered even being near that with the array, but I might have had no choice with rolling (assuming you roll like I do, straight down, no moving stats around).

But enough about me, let's talk about rolling...

One option that I rarely see used, but that I found really fun to use is the grid method of rolling stats. I use 3d6, but you're welcome to modify it as needed. Here's how it works.

Create a 3×3 grid. On the left side of the grid, mark each row with a physical stat: Strength, Dexterity, Constitution. At the top of the grid, mark each column with a mental stat: Intelligence, Wisdom, Charisma. Then roll 3d6 for each cell. Where it gets interesting is that you can only choose a number once.

$$\begin{array}{r|ccc} & \text{Int} & \text{Wis} & \text{Cha} \\ \hline \text{Str} & 16 & 8 & 13 \\ \text{Dex} & 5 & 16 & 6 \\ \text{Con} & 9 & 10 & 11 \end{array}$$

In this example, I can choose a 16 for Strength or Intelligence. But once I choose the 16 for Strength, my remaining options for Intelligence are 5 and 9. Were I to choose the 16 for Intelligence, the remaining options for Strength are 8 and 13.

Anyway, I like the mini-game aspect of rolling stats this way and thought I would share it in case some folks hadn't seen it before.

• This is innovative. It took me a while to get a feel for what you are doing with the grid, but I really like it. – KorvinStarmast Mar 8 '18 at 13:17

It's better to roll as that gives you, the player, more inspiration for the character. The possibility of having to take a character with an 18 and a couple of 6's is one that sparks many fun interpretations of the numbers. Using set arrays or even point buy systems leads to dull "optimized" characters that are often devoid of any quirks.

In AD&D, the suggestion is that the DM allow re-rolling if the character has less than two 15s, but I generally find that unnecessary with 4d6High3. Merl's rules obviously have a different "tipping point" of what is a hopeless character but that doesn't change the basic point which is that randomness gives a player a surprise from time to time and that's a great aid to roleplaying.

# Rolling the dice has better odds, but it’s risky and unfair.

You can use an AnyDice script (see below) to analyze the relationship between rolling dice and using the standard scores. If you sort the scores in order from highest to lowest, here’s the most common value for each ability, how likely you are to roll at least that well, and the average value (with standard deviation).

• 16 or higher 57% of the time, average 15.66 (σ=1.43)
• 14 or higher 69% of the time, average 14.17 (σ=1.44)
• 13 or higher 63% of the time, average 12.96 (σ=1.46)
• 12 or higher 58% of the time, average 11.67 (σ=1.53)
• 11 or higher 49% of the time, average 10.41 (σ=1.66)
• 9 or higher 51% of the time, average 8.50 (σ=1.95)

If the standard array used these values, it wouldn’t be as good of a deal as taking “average” hit points instead of rolling them, but it would very closely match what you are likely to roll, making a fair tradeoff between luck and reliability.

Unfortunately, the standard array uses lower values for the best and worst scores, which are exactly what players are most sensitive to. Players rolling fair dice will beat a high 15 more than half the time, and they’ll beat a low 8 more than half the time. Once you consider the thrill of gambling and the likelihood that players will simply ignore “unplayable” scores, I think you’ll find that rolling the dice greatly favors the players.

To put it another way, the standard array is equivalent to rolling slightly crappy but playable scores. That’s a problem if some players roll dice and others use the standard array. Unless everyone rolls poorly, with no do-overs, then you will run into the fairness problems that the standard array is intended to avoid. Therefore, I recommend that the players should decide to all roll dice or all use the standard array, as a group rather than choosing individually. If anybody wants to roll dice, it’s unfair to anyone who chooses the standard array, unless they fully understand the imbalance and deliberately choose it anyway to minimize risk.

Here’s the AnyDice script I used to obtain the statistics above.

ROLL: 6d[highest 3 of 4d6]
ARRAY: {15, 14, 13, 12, 10, 8}

loop N over {1..6} {
A: N@ARRAY
output N@ROLL named "Array [A]"
}


You can replace N@ROLL with N@ROLL - A to directly see how likely each ability is to fall above or below the array.

For another way of looking at this, you can extend the ability score point cost table (Basic Rules, p. 8) to evaluate rolled ability scores. Here’s another AnyDice script that extends the table two different ways:

COST: {0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 7, 9, 9, 9, 9}
COSTX: {-4, -3, -3, -2, -2, -1, -1, 0, 1, 2, 3, 4, 5, 7, 9, 11, 13, 15}
function: cost of SCORE:n { result: SCORE@COST }
function: extended cost of SCORE:n { result: SCORE@COSTX }

ROLLS: 6d[cost of [highest 3 of 4d6]]
ROLLX: 6d[extended cost of [highest 3 of 4d6]]

output ROLLS named "Cost"
output [highest of 0 and ROLLX] named "Cost+"


This shows that average value of rolled scores is about 28–30 points, depending on how much you value scores over 15, and you’re much more likely to roll above 27 points (55-60% chance) than you are to roll under 27 points (35–40%).

All that said, rolling terrible ability scores is not uncommon. If you roll up twenty characters, chances are that one of them will have no ability score higher than a 12 or 13. It’s very easy to roll up a character worth 20 points or less – that’s just over one standard deviation from average. If your whole groups rolls, somebody will lose (or cheat). I personally wish the designers had made the ability score tradeoff more like the hit points tradeoff, and set the standard array slightly higher than the average roll, rather than slightly lower.

The point-buy system is better than the standard array or rolling for stats because it lets you fine-tune your abilities. (The standard array costs 27 points in the point-buy system, which is exactly the maximum.)

The sum of your ability scores is not a useful measure of how "good" your stats are. For example, (a) DEX=16 STR=8 INT=8 is better for a rogue using finesse and ranged weapons than (b) DEX=14 STR=11 INT=11, even though the sum of a is 4 less than the sum of b. That is because (1) a rogue uses dexterity far more often, and for more important things, than strength and intelligence combined; and (2) two points are outright wasted in b, because STR=11 INT=11 has the same modifiers (STR=+0 INT=+0) as STR=10 INT=10. It is worth sacrificing a few points from the sum of your ability scores in order to ensure that the points you do have will be put to good use.

This is the cost of ability scores in point-buy:

ability  cost
8        0
9        1
10       2
11       3
12       4
13       5
14       7
15       9


Note that the cost sharply increases from 13 to 14, and then again from 14 to 15. If the sum of ability scores were paramount, then there would be no reason to buy any 14s or 15s. But I gladly take some 15s and 14s, even if it means 8s and 10s in two or three other abilities, in order to make my most important ability scores as high as they can be at level 1 and to minimize the amount of ability score increases needed to get them to 20. Fewer ASIs = more feats.

For example, for a Half-Elf rogue, I choose DEX=15, WIS=15, CHA=14, CON=10, STR=8, INT=8, and then apply the racial bonuses to get DEX=16, WIS=16, CHA=16, CON=10, STR=8, INT=8. That way, my three most important ability score modifiers are as high as they can possibly be at level 1, and all my ability scores are even numbers. (I say "modifiers" because what matters is your ability modifier, not the score itself. I could technically buy CHA=15 and then get CHA=17 from the Half-Elf's racial bonus, but I would still only have a CHA modifier of +3, and so I would still need to wait until level 4 to increase it to +4.)

I would not have the chance to fine-tune my ability scores that effectively if I rolled for the values. The only benefit of rolling would be the 57% chance of rolling one or more scores at least 16, which would enable me to start the game with a modifier of +4 (given a racial bonus), improving my early game experience and freeing up another ASI for a feat.

If a player wants to start with a +4 modifier, then maybe a good house-rule compromise is to allow a starting ability score of 16 to be bought for 11 points. I know I would take that option.

Both have their merits.

The average array prevents overpowered characters from dominating a campaign while ensuring that players will get a fair spread of attributes across their point values, while the appeal of Random Dice Rolling means that a character's abilities will be far more varied and interesting.

It really depends on how you want to play the game - if you're most interested in keeping balance among players (A good idea if players you have are prone to power-gaming, or if they are new players) then a point spread is better. For more variety and for a little more fun (players who you can trust, or who enjoy the challenge of a random character) you can give the random rolls a chance, and offer re-rolls to help balance out the possiblity of a bad roll.

Of course...there's an alternative to both of these that I like...

I've been in a number of successful campaigns that have used the point buy system in leiu of random-rolling or point spread. It's fairly simple once you get used to it, and allows character customization while encouraging balance at the same time.

Very simply put - you give players a set number of points, and they use these points to purchase thier abilities. It's a little different in 5e, so I suggest checking it out for yourself in the BD&D on page 9, which can be found for free.

• It's worth noting that in 5e the maximum allowed score with point buy is 15, you get 27 points, and the table for allocating them is on page 9 of BD&D. – wax eagle Sep 23 '14 at 19:53

Statistically, the two options are pretty much equivalent. Based on the chart Salvador posted above, it appears that they did the math themselves, and decided that the threshold to 'round up' for those values was somewhere around .75, vs. the standard .5.

Rolling gives you the opportunity to get much better (or much worse) values than the 'Joe Hero' package, and also gives you the opportunity to end up with much 'flatter' or 'spikier' result sets as well. You could (conceivably) end up with: 18, 3, 3, 3, 3, 3; or 10, 10, 10, 10, 10, 10.

Over a large enough set of rolls, you'd end up with averages right around the 'Joe Hero' package, but rolling enables a much wider set of results. The choice is yours (or your DM's) to make.

Rolling stats is the only way you can end up starting with 18-20 in a stat. Of course, it's also the only way you can end up starting with a 3-7 in a stat, too, so keep that in mind when you make your decision.

Personally, I prefer the pre-set array, because it makes things easier for newer players, and guarantees a relatively even playing field within a group.

As a compromise (sort of) you can allow players to roll to determine their array. The player would roll 3d6 to determine a random array that they have to use.

3d6     Array
3       14  13  13  12  12  10
4       14  14  13  12  10  10
5       14  14  13  11  11  10
6-7     15  13  12  12  11  10
8-9     15  13  13  12  10  10
10-11   15  14  13  12  10   9
12-13   15  14  13  12  10   8
14-15   15  14  14  11   9   8
16      15  15  12  10  10   9
17      15  15  13  10   9   9
18      15  15  14   9   9   8


Alternately, if you wanted a tighter range of arrays, the players could roll 1d12+4, 1d10+5, 1d8+6, or 1d6+7 Players who get the 'lower' stats have a higher average bonus, making them more jack of all trades types, while players with a few high stats have a lower average.

In my experience, rolling is more fun, but it has to be clipped. Our house rule is that if nothing is over 13 or more than 2 are 16 or greater, you need to reroll. Also, either everyone rolls, or nobody does. The advantages are that it keeps the table min/maxers somewhat in check and results in interesting situations that you otherwise wouldn't have. I'd also include rolling for HP if you're rolling for stats. If the goal is to spice things up with the capricious will of the RNG, go whole hog.

For example, in my last campaign, we had a rogue who started with 20 DEX, but I think 9 CON and 7 STR. Combined with some unfortunate HP rolls, he ended up being a glass cannon for whom no lock or trap was too tough. He was too weak to carry a full thieves' pack, and his head was eventually smashed in a single blow by a revenant at level 6 or so. Definitely not the kind of character you'd have with the standard array.