Ability Score Generation [E6]

[Darkxarth]

Hello all,

I am currently working on an E6 variant project and am looking for different ways to generate ability scores. I have 3 NPC arrays, 3 PC arrays, and 3 PC rolling methods. I just want to make sure that (i) the PC arrays and rolls are generally superior to the NPC arrays and (ii) the PC arrays and rolling methods are approximately equivalent. As an English major, I have taken one math class in the last 4 years, and it was not probability and statistics. So, I turn to you, Playgrounders, to help me with the number-crunching of ability score generation.

NPC Arrays

Common Array: 9, 10, 10, 11, 11, 12
Heroic Array: 8, 10, 11, 12, 13, 14
Paragon Array: 8, 9, 10, 12, 14, 16


PC Arrays

Basic Array: 8, 10, 13, 14, 15, 16
Specialist Array: 8, 10, 11, 12, 14, 18
Generalist Array: 12, 13, 13, 14, 14, 15


PC Rolling Methods

Standard Method: 4d6, reroll any 1s and drop the lowest die.
Average Method: 3d6 + 4, drop the lowest die.
Heroic Method: 5d6, drop the lowest two dice.


Mostly, it's just the rolling methods with which I am having trouble. I can't determine the value of "reroll 1s" compared to "+4" compared to extra dice.

Your assistance is greatly appreciated and suggestions are welcome.

[Yora]

What is it you really want to know here?

[Patrikc]

What is it you really want to know here?

Did you read his post?


I just want to make sure that (i) the PC arrays and rolls are generally superior to the NPC arrays and (ii) the PC arrays and rolling methods are approximately equivalent.



The value of "reroll 1s" is simply that instead of 1-6 you use 2-6, +4 is .5 points higher than an average d6 roll, but obviously takes out the variables. On average these two are equal.

Until my math classes in September, that's about as crunching as my numbers will go, though.

[Darkxarth]

What is it you really want to know here?

I am trying to ensure:

PC arrays are greater than NPC arrays
PC arrays are approximately equal to each other
Rolling methods are approximately equal to each other
Rolling methods are approximately equal to arrays (least of my concerns)



Until my math classes in September, that's about as crunching as my numbers will go, though.
Ah well, thanks.

[Khellendross]

What is E6?

Nvm, googled it. Seems cool.

[Yora]

E6 (http://www.enworld.org/forum/general-rpg-discussion/206323-e6-game-inside-d-d.html) is a variant in which you gain level normaly until 6th, and then you just get a feat for every additional 5000 XP. Removes all 4th level and higher spells from the game as well as high level NPCs that can just save the world every other day, just by being high level. The result is that you end up with rather "normal guys" as PCs and what they do actually matters, becausenobody there are no epic spellcasters who could do it instead.

---

If you want approximately equal arrays, just add up the ability modifiers:
Common Array: 9, 10, 10, 11, 11, 12 = -1 +0 +0 +0 +0 +1 = +0
Heroic Array: 8, 10, 11, 12, 13, 14 = -1 +0 +0 +1 +1 +2 = +3
Paragon Array: 8, 9, 10, 12, 14, 16 = -1 -1 +0 +1 +2 +3 = +4

PC Arrays
Basic Array: 8, 10, 13, 14, 15, 16 = -1 +0 +1 +2 +2 +3 = +7
Specialist Array: 8, 10, 11, 12, 14, 18 = -1 +0 +0 +1 +2 +4 = +6
Generalist Array: 12, 13, 13, 14, 14, 15 = +1 +1 +1 +2 +2 +2 = +9

As you see, the Paragon array is only slightly better than the Heroic array. And the Specialist array is actually the weakest PC array. Also, the PC arrays are much more powerful than the NPC ones.

PC Rolling Methods
Standard Method: 4d6, reroll any 1s and drop the lowest die. =~ 13 13 14 14 14 14 = +10
Average Method: 3d6 + 4, drop the lowest die. =~ 14 14 14 15 15 15 = +12
Heroic Method: 5d6, drop the lowest two dice. =~ 15 15 15 16 16 16 = +15

It's been some time (5 months *cough*), but I think this is a somewhat decent approximation of the rolling methods. They are actually a bit too high, because I didn't consider the bell curve you get when rolling multiple dice, but I think doing that would be a calculation far to complex to make at the drop of a hat.

[Autopsibiofeeder]

I am trying to ensure:


Rolling methods are approximately equal to each other




Well, it depends on your definition of approximately, but I would say they are not quite. For instance, you can't get an 18 with 3d6+4, drop lowest die (unless you treat the +4 as a die): it maxes out at 16. This method, in a sense, artificially centers most scores a bit over 11.

Without making it needlessly complicated:
The standard method gives results of 6-18, the average gives method results of 6-16. Not only is the spread larger for the former, the average score will be a bit higher as well (more or less one point per score, on average). This is assuming that the dropping of the lowest die has the same effect for both methods (which it does not, but this is the not-complicated version).

The heroic method yields results of 3-18, with the chance of very low scores being relatively small (compared to high scores). Still, this method will result in the most diverse/extreme sets of scores where most scores will be fairly high (13-14 and up) but every set is likely to end up with one or two low scores (8 or even lower).

[Khellendross]

E6 (http://www.enworld.org/forum/general-rpg-discussion/206323-e6-game-inside-d-d.html) is a variant in which you gain level normaly until 6th, and then you just get a feat for every additional 5000 XP. Removes all 4th level and higher spells from the game as well as high level NPCs that can just save the world every other day, just by being high level. The result is that you end up with rather "normal guys" as PCs and what they do actually matters, becausenobody there are no epic spellcasters who could do it instead.

---

If you want approximately equal arrays, just add up the ability modifiers:
Common Array: 9, 10, 10, 11, 11, 12 = -1 +0 +0 +0 +0 +1 = +0
Heroic Array: 8, 10, 11, 12, 13, 14 = -1 +0 +0 +1 +1 +2 = +3
Paragon Array: 8, 9, 10, 12, 14, 16 = -1 -1 +0 +1 +2 +3 = +4

PC Arrays
Basic Array: 8, 10, 13, 14, 15, 16 = -1 +0 +1 +2 +2 +3 = +7
Specialist Array: 8, 10, 11, 12, 14, 18 = -1 +0 +0 +1 +2 +4 = +6
Generalist Array: 12, 13, 13, 14, 14, 15 = +1 +1 +1 +2 +2 +2 = +9

As you see, the Paragon array is only slightly better than the Heroic array. And the Specialist array is actually the weakest PC array. Also, the PC arrays are much more powerful than the NPC ones.

PC Rolling Methods
Standard Method: 4d6, reroll any 1s and drop the lowest die. =~ 13 13 14 14 14 14 = +10
Average Method: 3d6 + 4, drop the lowest die. =~ 14 14 14 15 15 15 = +12
Heroic Method: 5d6, drop the lowest two dice. =~ 15 15 15 16 16 16 = +15

It's been some time (5 months *cough*), but I think this is a somewhat decent approximation of the rolling methods. They are actually a bit too high, because I didn't consider the bell curve you get when rolling multiple dice, but I think doing that would be a calculation far to complex to make at the drop of a hat.


I would love to play in an E6 game after reading it. I've always wanted to play a wizard but have never gotten the chance. Either teir 1 isn't allowed or I'm the party cleric as normal. Sad thing is I'd just be a Abjuration specialist and not go for pure power just a nice concept like a white robed wizard from Dragonlance.

[Holocron Coder]

This (http://rumkin.com/reference/dnd/diestats.php) is what you want.

[Darkxarth]

If you want approximately equal arrays, just add up the ability modifiers:
Common Array: 9, 10, 10, 11, 11, 12 = -1 +0 +0 +0 +0 +1 = +0
Heroic Array: 8, 10, 11, 12, 13, 14 = -1 +0 +0 +1 +1 +2 = +3
Paragon Array: 8, 9, 10, 12, 14, 16 = -1 -1 +0 +1 +2 +3 = +4

PC Arrays
Basic Array: 8, 10, 13, 14, 15, 16 = -1 +0 +1 +2 +2 +3 = +7
Specialist Array: 8, 10, 11, 12, 14, 18 = -1 +0 +0 +1 +2 +4 = +6
Generalist Array: 12, 13, 13, 14, 14, 15 = +1 +1 +1 +2 +2 +2 = +9

As you see, the Paragon array is only slightly better than the Heroic array. And the Specialist array is actually the weakest PC array. Also, the PC arrays are much more powerful than the NPC ones.

Thank you very much. It did not even occur to me to add up the modifiers and compare them. It looks like I will need to take another look at the PC arrays and try to sort them out.


Well, it depends on your definition of approximately, but I would say they are not quite. For instance, you can't get an 18 with 3d6+4, drop lowest die (unless you treat the +4 as a die): it maxes out at 16. This method, in a sense, artificially centers most scores a bit over 11.

Without making it needlessly complicated:
The standard method gives results of 6-18, the average gives method results of 6-16. Not only is the spread larger for the former, the average score will be a bit higher as well (more or less one point per score, on average). This is assuming that the dropping of the lowest die has the same effect for both methods (which it does not, but this is the not-complicated version).

The heroic method yields results of 3-18, with the chance of very low scores being relatively small (compared to high scores). Still, this method will result in the most diverse/extreme sets of scores where most scores will be fairly high (13-14 and up) but every set is likely to end up with one or two low scores (8 or even lower).

The Average Method is designed to help prevent low ability scores, but does not allow for starting scores above 16, either. Heroic Method appears to be doing exactly what I want, which is a random range between 3-18, centered around the +1/+2 modifier mark.


This (http://rumkin.com/reference/dnd/diestats.php) is what you want.
Aha! Perfect for those complicated methods.

Thanks to everyone for your help.

[Autopsibiofeeder]

The Average Method is designed to help prevent low ability scores, but does not allow for starting scores above 16, either.

But the thing is, it doesn't prevent low scores. There's a 10,7% chance to get a negative modifier with 3d6+4, drop lowest (average method) opposed to a 5.8% chance for a negative modifier with 4d6, reroll 1 and drop lowest. You are only 'trimming' the upper range of outcomes and enhances the probability of low scores.

The 5d6 method has, as I mentioned, also a fairly large chance that some scores turn out low(8%), but focuses most scores around 14.

[Murdim]

PC Rolling Methods

Standard Method: 4d6, reroll any 1s and drop the lowest die.
Average Method: 3d6 + 4, drop the lowest die.
Heroic Method: 5d6, drop the lowest two dice.


Solving probability problems through the shameless abuse of spreadsheets. Because intellectual laziness is fun ! :smallbiggrin:

Standard Method
Sample space size : 625 (5^4) - outcomes with a 1 are just ignored
Mean : 13,43
Probability of a 6 : 0,16% (1 outcome)
Probability of a 7 : 0,64% (4 outcomes)
Probability of a 8 : 1,60% (10 outcomes)
Probability of a 9 : 3,36% (21 outcomes)
Probability of a 10 : 6,08% (38 outcomes)
Probability of a 11 : 9,28% (58 outcomes)
Probability of a 12 : 12,64% (79 outcomes)
Probability of a 13 : 15,04% (94 outcomes)
Probability of a 14 : 16,00% (100 outcomes)
Probability of a 15 : 14,56% (91 outcomes)
Probability of a 16 : 11,20% (70 outcomes)
Probability of a 17 : 6,72% (42 outcomes)
Probability of a 18 : 2,72% (17 outcomes)

Average method
Sample space size : 216 (6^3)
Mean : 12,5
Probability of a 6 : 0,46% (1 outcome)
Probability of a 7 : 1,39% (3 outcomes)
Probability of a 8 : 3,24% (7 outcomes)
Probability of a 9 : 5,56% (12 outcomes)
Probability of a 10 : 8,80% (19 outcomes)
Probability of a 11 : 12,50% (27 outcomes)
Probability of a 12 : 15,74% (34 outcomes)
Probability of a 13 : 16,67% (36 outcomes)
Probability of a 14 : 15,28% (33 outcomes)
Probability of a 15 : 11,57% (25 outcomes)
Probability of a 16 : 6,94% (15 outcomes)
Probability of a 17 : 1,39% (3 outcomes)
Probability of a 18 : 0,46% (1 outcome)

Heroic Method:
Sample space size : 7776 (6^5)
Mean : 13,43
Probability of a 3 : 0,01% (1 outcome)
Probability of a 4 : 0,06% (5 outcomes)
Probability of a 5 : 0,19% (15 outcomes)
Probability of a 6 : 0,53% (41 outcomes)
Probability of a 7 : 1,16% (90 outcomes)
Probability of a 8 : 2,19% (170 outcomes)
Probability of a 9 : 3,81% (296 outcomes)
Probability of a 10 : 6,04% (470 outcomes)
Probability of a 11 : 8,55% (665 outcomes)
Probability of a 12 : 11,33% (881 outcomes)
Probability of a 13 : 13,57% (1055 outcomes)
Probability of a 14 : 14,85% (1155 outcomes)
Probability of a 15 : 14,29% (1111 outcomes)
Probability of a 16 : 12,02% (935 outcomes)
Probability of a 17 : 7,84% (610 outcomes)
Probability of a 18 : 3,55% (276 outcomes)

Some observations :

_ Surprisingly, the Standard and Heroic method have the same mean - well, almost the same, actually, but the level of approximation is more than acceptable. The main difference is that the Heroic method involves a higher degree of dispersion : it is more likely to result in very high or very low scores, while the Standard method is more likely to result in scores closer to the average - here, between 12 and 16. Also, the Heroic method has a lower minimum score than the Standard method.

_ A possible "fix" to put the Average method on par with the two others is to enhance the "free die" from an automatic 4 to an automatic 5, which puts the mean score to 13,46 (0,03 above the two others). The minimum value is improved from 6 to 7, it is even less dispersed than the Standard method, and you're extremely unlikely to get a 18 or a value lower than 8 (i.e a 7).

Average method Mk.2
Sample space size : 216 (6^3)
Mean : 13,46
Probability of a 7 : 0,46% (1 outcome)
Probability of a 8 : 1,39% (3 outcomes)
Probability of a 9 : 3,24% (7 outcomes)
Probability of a 10 : 5,56% (12 outcomes)
Probability of a 11 : 8,80% (19 outcomes)
Probability of a 12 : 12,50% (27 outcomes)
Probability of a 13 : 15,74% (34 outcomes)
Probability of a 14 : 16,67% (36 outcomes)
Probability of a 15 : 15,74% (34 outcomes)
Probability of a 16 : 12,50% (27 outcomes)
Probability of a 17 : 6,94% (15 outcomes)
Probability of a 18 : 0,46% (1 outcome)

EDIT : For the Average method, I assumed that the 4 could be discarded like any other die, else it would give an even worse mean score. With a 5, it doesn't really matter anymore. There's only one outcome out of 216 - the triple 6 - where it would matter at all, and downgrading the only "18" outcome into a "17" doesn't have a meaningful impact on the mean value.

[Yora]

When creating arrays, I would use the point buy method.

In my E6 game I use:

NPC-Class NPCs: Point Buy 15
PC-Class NPCs: Point Buy 20
PCs: Point Buy 25

Point Buy 15 allows a character to have 11, 11, 11, 10, 10, 10, which is (implicitly) defined as THE standard array for human NPCs.
Point Buy 25 is Standard Point Buy and allows for the Heroic Array of 15, 14, 13, 12, 10, 8.
The big advantage of Point Buy over arrays is, that a player can chose how to arrange his scores. Some classes need one ability really high, while others are better off with having "good" scores in several. When you plan to multiclass its even more usefull to make up the ability scores as you like.
Arrays are useful as examples how you can arrange ability scores with a given PB limit. For example, I almost always make PB 25 characters with 15, 14, 13, 12, 10, 8 or 14, 14, 13, 12, 10, 10 and only very rarely use something else.

However, some people don't like to give players complete control how to assign ability scores and force them to work with scores that are not really optimal for their concept. In this case a fixed array would bring the limitations of a die roll, while providing the security of having at least an average set of scores. Rolling is more of a gamble, as you can get both very high and very low rolls.

To come up with one, or more, method to generate ability scores for your campaign, you should first think about how you want ability generation to be: Force the players to work with what they got? Allow for chance for both high and low abilities? Make all characters roughly equal in ability scores, or very different?
3d6 is the most extreme, Point Buy the most generous.

[jokey665]

Thank you very much. It did not even occur to me to add up the modifiers and compare them. It looks like I will need to take another look at the PC arrays and try to sort them out.
Well, while that's one way of doing it, it's not necessarily the best. Here's the point buy of each of the builds:

NPC
Common 16
Heroic 20
Paragon 23

PC
Basic 31
Specialist 31
Generalist 34

[Darkxarth]

Solving probability problems through the shameless abuse of spreadsheets. Because intellectual laziness is fun ! :smallbiggrin:

Some observations :

_ Surprisingly, the Standard and Heroic method have the same mean - well, almost the same, actually, but the level of approximation is more than acceptable. The main difference is that the Heroic method involves a higher degree of dispersion : it is more likely to result in very high or very low scores, while the Standard method is more likely to result in scores closer to the average - here, between 12 and 16. Also, the Heroic method has a lower minimum score than the Standard method.

_ A possible "fix" to put the Average method on par with the two others is to enhance the "free die" from an automatic 4 to an automatic 5, which puts the mean score to 13,46 (0,03 above the two others). The minimum value is improved from 6 to 7, it is even less dispersed than the Standard method, and you're extremely unlikely to get a 18 or a value lower than 8 (i.e a 7).
Hmm, that seems to work well as a fix to the Average Method. They still cannot get an 18 in any score, but the minimum is now 7. Maybe +6 would be better, or would that be too much (I think so).

[Murdim]

Hmm, that seems to work well as a fix to the Average Method. They still cannot get an 18 in any score, but the minimum is now 7. Maybe +6 would be better, or would that be too much (I think so).
Yeah, that would be too much. With +5, the Average method is already slightly more advantageous in theory than the two others.