Graphing point buy vs. dice comparisons for stat generation


Pathfinder First Edition General Discussion

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It seems several older threads have partially addressed my question of how different dice methods compare to point buy -- but typically only address the point buy equivalency of the average array created by a certain dice roll method. Several posters mentioned the problem of variance from that mean, but without stats. I figured I'd try to fill some of that in.

Method (via Python scripting):
* Generate a large number of character stat arrays via the chosen die roll method.
* Calculate the point-buy cost of each of those rolled arrays (discarding any with rolled stats less than 7, since you can't point buy that low)
* Determine the mean and standard deviation of that set of point-buy costs
* Most fun, graph the distribution of point-buy costs.

Conclusion: I've long been a hold-out on point buy, thanks to my 1e/2e habits, despite knowing the variance issue. By visualizing just how skewed the characters in a party can be, though, I think I've convinced myself (as both GM and player) to make the switch.

3d6: of 10,000 stat arrays rolled, 4,444* were dropped because they had at least one score below 7. The remaining arrays had a mean of 8.04 points and standard deviation of 8.80 points. That is, about 2/3 of characters rolled via 3d6 fall somewhere between a -1 point buy and a 17 point buy.

...But we also find characters as bad as a -18 point buy, or as good as 46 points.

Histogram.:

-18 : *
-17 : *
-16 : *
-15 : *
-14 : *
-13 : *
-12 : *
-11 : **
-10 : ****
-9 : *****
-8 : *******
-7 : *******
-6 : ************
-5 : **************
-4 : *******************
-3 : **********************
-2 : ************************
-1 : ****************************
0 : **********************************
1 : **************************************
2 : ***************************************
3 : ****************************************
4 : ***********************************************
5 : **********************************************
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23 : *********
24 : *********
25 : *******
26 : *****
27 : ****
28 : *******
29 : ****
30 : ***
31 : ***
32 : **
33 : ***
34 : **
35 : **
36 : *
37 : *
38 : *
39 : *
40 : *
41 : *
42 : *
43 : *
46 : *

4d6, drop lowest gives a mean point buy cost of 20.5, and standard deviation of 10.6, with only 1,543 of 10,000 arrays discarded for sub-7 scores. So 2/3 of characters rolled this way fall between a 10-point buy and a 31-point buy, but we also see characters as low as -10 points and as high as 64 among our 10,000.

Histogram.:

-10 : *
-9 : *
-7 : *
-5 : *
-4 : **
-3 : **
-2 : *****
-1 : *****
0 : ******
1 : *********
2 : **********
3 : ***********
4 : **************
5 : ******************
6 : *********************
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45 : ****
46 : ****
47 : **
48 : **
49 : ***
50 : **
51 : **
52 : *
53 : *
54 : *
55 : *
56 : *
57 : *
58 : *
59 : *
60 : *
61 : *
62 : *
64 : *

Note that the tails on these graphs get even longer if I do 100,000 or 1 million samples, because in a large enough pool you eventually hit the all-7s and all-18s characters, and the curves get smoother with larger numbers, but the descriptive stats don't change much.

Other rolling methods that want a graph?


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Here's 2d6+6: 0 of 10,000 dropped for sub-7 scores, mean of 25.7 points, standard deviation of 10.5 points. In 10k characters, we range from a -1 point-buy array to a 74 point-buy array.

Histogram:

-1 : *
0 : *
1 : **
2 : **
3 : ***
4 : ****
5 : *****
6 : ******
7 : ********
8 : *************
9 : *****************
10 : *****************
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45 : ***********
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49 : ******
50 : ****
51 : ****
52 : *****
53 : **
54 : **
55 : *
56 : **
57 : **
58 : *
59 : **
60 : *
61 : *
62 : *
63 : *
65 : *
66 : *
68 : *
69 : *
74 : *


Thanks. I was just thinking of this the other day.


Wow! Great work Murph.! I've always preferred point buy myself since I was first introduced to it, but it's nice to be able to visualize the numbers on rolling.

As to other methods you might try, re-rolling ones is a common house rule. It would probably be pointless to reexamine 2d6+6, since it never generates sub-7 scores, but it would be interesting to see how far it brings 3d6 up.

I've also heard 10+2d4, but that generates some pretty ridiculous high-powered characters, so testing it might be a bit superfluous.


7d6 drop three highest, 8d6 drop three highest.


Awesome, I love this sort of stuff.

I think if you start with str and work your way through to cha using any one of these methods and then pick a class based on what you have rolled you'd get a better feel for your character than with point buy. For extra hard core you could pick races before rolling.


Lurk3r wrote:

It would probably be pointless to reexamine 2d6+6, since it never generates sub-7 scores, but it would be interesting to see how far it brings 3d6 up.

I've also heard 10+2d4, but that generates some pretty ridiculous high-powered characters, so testing it might be a bit superfluous.

2d6+6 is in my first comment after OP. Very similar curve to 4d6, but slid 5 points more powerful.

2d4+10, yes, is pretty nuts. Single-character campaign fodder, that. Mean = 46.6 points, std dev = 9.8 points, with spread of 17-88 points in 10k stat arrays.

2d4+10 histogram:

17 : *
19 : *
20 : *
21 : *
22 : *
23 : ***
24 : **
25 : ***
26 : *****
27 : *****
28 : *******
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72 : *
73 : **
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75 : *
76 : *
77 : **
78 : *
79 : *
80 : *
81 : *
82 : *
83 : *
84 : *
88 : *


Looking at that 4d6 graph drives home the main issue I have with rolling.
I mean, might as well tell people to roll 3d6 and then add 10 and that's how many points you get in point buy.

The only random methods I've seen that alleviate this issue are the ones whose rolls are altered as you go based on your results.
The playing cards method is the simplest I've seen.


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All this has done is convince me that point buy is the only good method for character generation.


Glutton wrote:
7d6 drop three highest, 8d6 drop three highest.

7d6, drop 3 highest creates truly mediocre characters, but with occasional standouts when you roll handfuls of 4+. If for some reason you want to roll stats for every NPC in a farming village (including that one powerful druid who lives out in the sticks), this method might be for you.

Mean = 2.7 points, std dev = 8.4 points, range -21 - 49. 6001 / 10000 arrays had to be discarded for stats under 7; 224 for stats over 18.

7d6, drop 3 highest histogram:

-21 : *
-20 : *
-19 : *
-18 : *
-17 : *
-16 : **
-15 : **
-14 : ***
-13 : ***
-12 : *******
-11 : *********
-10 : ************
-9 : ************
-8 : ********************
-7 : ***********************
-6 : *****************************
-5 : **************************
-4 : **************************
-3 : *******************************
-2 : **********************************
-1 : ******************************************
0 : ***************************************
1 : ********************************************
2 : *************************************
3 : ***********************************
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20 : *****
21 : ****
22 : ***
23 : ****
24 : **
25 : **
26 : *
27 : **
28 : **
29 : *
30 : *
31 : *
32 : *
33 : *
34 : *
35 : *
37 : *
38 : *
44 : *
45 : *
49 : *

8d6, drop 3 highest looks like basically a more time-consuming way to roll 4d6, drop low: you have to throw out twice as many rolled arrays for stats out of allowed point-buy range (7-18); after that, the descriptive stats are very similar. (Of course, if you didn't limit yourself to the 7-18 range, this method would be much, much, much better.)

Mean = 20.8 points, std dev = 11.9. Range -14 to 77, 867 / 10000 discarded for stats under 7, 2872 for stats over 18.

8d6, drop 3 highest histogram:

-14 : *
-10 : *
-9 : *
-8 : *
-7 : **
-6 : ***
-5 : ***
-4 : **
-3 : ***
-2 : ******
-1 : *******
0 : **********
1 : ************
2 : *************
3 : ************
4 : *********************
5 : *******************
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51 : ***
52 : **
53 : **
54 : ***
55 : **
56 : **
57 : *
58 : *
59 : *
60 : *
61 : *
63 : *
64 : *
67 : *
69 : *
70 : *
77 : *

Dark Archive

The problem with rerolling scores less than 7 is it really pushes up your average. It also takes away an element of risk from the randomness.

Average point buys for different random stat generation methods.


While its interesting to see, i dont think its very useful to actually evaluate theusefulness ofstats regenerated by rolleddice; point buy is by no means a perfect balanced system, quite the opposite, and so saying you rolled stats equal to a 23 pb does not mean you will actually have _better_ stats than a 15 pb char.


Pryllin wrote:

The problem with rerolling scores less than 7 is it really pushes up your average. It also takes away an element of risk from the randomness.

Average point buys for different random stat generation methods.

Your linked post was one of the ones I found when considering whether to crop the scores to 7-18. I liked your extrapolations, but figured I'd stick to the "canonical" point buy range.

I think there's still a fair amount of "risk" in rounding / re-rolling scores below 7 -- that's kind of the point of the distribution graphs -- and the groups I play in that roll random scores tend to allow pity re-rolls anyways if somebody comes out with a 5.

Dark Archive

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There's always risk with random dice rolls, but many people think that risk lies in rolling poor stats, in which case rerolls may or may not be allowed. Risk also lies in someone getting high rolls and dominating the party with awesome stats.

The simple fact of the matter is that using a random method, someone will roll better than the rest of the party, and someone will roll worse. If you're going to give rerolls to try and give everyone similar stat values, you may as well just cut to the chase and assign a point buy method.


How about the Way of the Wicked stat gen of 1d10+7?

cheers


5d4-2?


I would be interested in finding methods of rolling ability scores that narrow the standard deviation, thereby reducing the chance of such a wide disparity between player characters.

Dungeons and Dragons 3.5 had the Rerolling rule (p.8 PHB) that effectively narrowed the range of ability score arrays; however that rule did not make it into Pathfinder.

I have toyed with a system called the Dice Point method, wherein you get 5 dice points to spend on your ability scores. Each dice point equals a 6 on the die, so if you choose to put 1 into Strength you only roll two dice and then add 6. Before you roll you distribute the 5 dice points among your stats as you wish, but you can only spend up to 3 in any one stat. If you spend 3 dice points in one stat, then that stat automatically equals 18.

I have found that this method reduces the standard deviation a little.


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4d6 27/26/25 'mirror method'

- Roll 4d6, select the three highest. This is the first score.
Subtract the first score from 27. This is the second score.
- Roll 4d6, select the three highest. This is the third score.
Subtract the third score from 25. This is the fourth score.
- Roll 4d6, select the three highest. This is the fifth score.
Subtract the fifth score from 23. This is the sixth score.

Arrange the scores to taste

Everyone who uses this method ends up with an ability score total of 77.


Couple more --

1d10+7 ensures all the stats fall in 7-18 (or 8-17, more precisely), but actually has a wider standard deviation of point buy values than many of the others, because there's less clustering of individual scores around a central value. 0 / 10,000 arrays discarded, mean 22.9 points, std dev 11.4 points.

histogram:

-9 : *
-7 : *
-6 : *
-5 : *
-4 : ***
-3 : **
-2 : ****
-1 : ****
0 : ******
1 : ********
2 : *********
3 : ***********
4 : ***********
5 : **************
6 : ***************
7 : *********************
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69 : *

5d4-2: another "villager's" point buy option -- the average individual stat is going to be 8.5. 2,854 / 10,000 arrays dropped for sub-7 stats. Of remaining, mean 5.7 point buy, std dev 7.2 points.

histogram:

-14 : *
-13 : *
-12 : *
-11 : **
-10 : ***
-9 : ***
-8 : *******
-7 : *******
-6 : **********
-5 : ****************
-4 : *********************
-3 : ************************
-2 : ******************************
-1 : ***********************************
0 : ***************************************
1 : ***************************************
2 : ********************************************
3 : ***********************************************
4 : **********************************************
5 : *******************************************
6 : **********************************************
7 : ***********************************************
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23 : ***
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25 : **
26 : **
27 : **
28 : **
29 : *
30 : *
31 : *
32 : *
33 : *
34 : *
36 : *
41 : *

Just for fun, what if we generate stats by flipping a coin? 9d2 makes stats in the 9-18 range, meaning virtually no negative points. Fairly high / tight range: mean 26.9 point buy, std dev 6.7 points.

histogram:

8 : *
9 : *
10 : *
11 : **
12 : **
13 : ***
14 : *******
15 : ***********
16 : *************
17 : ******************
18 : *********************
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46 : **
47 : **
48 : *
49 : *
50 : *
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55 : *
56 : *
60 : *


I prefer the James Tiberius Kirk approach to stat generation:

"Never tell me the odds!"


As usual when comparing the charts miss the advantage point buy has over rolling, the greater utility of even ability scores over odd ability scores because most of the breakpoints for scores are at even numbers. Spells and magic items which affect ability scores do it by an even number because of this. A bard with a 15/13/11/8/11/17 (a 25 point character) is not 25% better than a 14/12/12/8/10/17 Bard (a 20 point character) and all other things being equal is somewhat inferior.


cnetarian wrote:
As usual when comparing the charts miss the advantage point buy has over rolling, the greater utility of even ability scores over odd ability scores because most of the breakpoints for scores are at even numbers. Spells and magic items which affect ability scores do it by an even number because of this. A bard with a 15/13/11/8/11/17 (a 25 point character) is not 25% better than a 14/12/12/8/10/17 Bard (a 20 point character) and all other things being equal is somewhat inferior.

More generally, it doesn't cover the advantage of being able to set the scores exactly as you want them.

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
Lakesidefantasy wrote:
I would be interested in finding methods of rolling ability scores that narrow the standard deviation, thereby reducing the chance of such a wide disparity between player characters.

Oh, that's easy.

"Best 4 dice of 20d4" is going to have a very small standard deviation.
That doesn't mean it's a particularly good way of rolling characters.


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JohnF wrote:
Lakesidefantasy wrote:
I would be interested in finding methods of rolling ability scores that narrow the standard deviation, thereby reducing the chance of such a wide disparity between player characters.

Oh, that's easy.

"Best 4 dice of 20d4" is going to have a very small standard deviation.
That doesn't mean it's a particularly good way of rolling characters.

Or you could go with 18d1. That would have a really tiny standard deviation.


JohnF wrote:
Lakesidefantasy wrote:
I would be interested in finding methods of rolling ability scores that narrow the standard deviation, thereby reducing the chance of such a wide disparity between player characters.

Oh, that's easy.

"Best 4 dice of 20d4" is going to have a very small standard deviation.
That doesn't mean it's a particularly good way of rolling characters.

Would that narrow the curve? If so, then I would be interested to see the standard deviation of the best three of 20d6.

My next question would be how much spread in the curve is acceptable and competitive with point buy? Or, how low does the total point buy need to go before a particular random rolling method is more desireable.


thejeff wrote:
cnetarian wrote:
As usual when comparing the charts miss the advantage point buy has over rolling, the greater utility of even ability scores over odd ability scores because most of the breakpoints for scores are at even numbers. Spells and magic items which affect ability scores do it by an even number because of this. A bard with a 15/13/11/8/11/17 (a 25 point character) is not 25% better than a 14/12/12/8/10/17 Bard (a 20 point character) and all other things being equal is somewhat inferior.
More generally, it doesn't cover the advantage of being able to set the scores exactly as you want them.

I don't really worry much about that, to tell the truth, maybe it is from spending too much time with old style make a character out of random rolls, but you are correct. A barbarian who rolled straight 15s would be a 42 point character but a decent 20 point build barbarian with a 17 strength and 16 constitution would be hitting more often and harder while raging longer.

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
Lakesidefantasy wrote:
I would be interested to see the standard deviation of the best three of 20d6.

A quick back-of-the-envelope calculation says, if I've got it right, that best 3 of 20d6 is going to result in a score of 18 around 2/3 of the time, so the standard deviation is going to be pretty small.


I ran a top three of 10d6 for 10,000 trials and got a point-buy equivalent standard deviation of around 11 points, a median of around 65 points, and a range of around 80 points.

That is worse than "top three" of 3d6 with a standard deviation of around 9 points, median of around 3 points, and range of around 100 points.

However, I ran a top three of 20d6 for 10,000 trials and got a standard deviation of around 7 points, but I think that is because the results got smashed up against the maximum of a 102 point-buy equivalent which reduced the overall range to around 50 points.

Adding dice does not seem to reduce the spread of the results. plus, the average equivalent point-buy skyrockets. The top three of 20d6 median point-buy equivalent was around 90 points!

Oh well, now that I have a new way of looking at this problem I'll keep searching for and testing score generation methods that satisfy my desire for randomness, customization and egalitarianism.

Perhaps range limitations could be the solution. That's the way Dungeons and Dragons 4.0 does it by limiting the total bonus modifiers to between 4 and 8.

I have a question for the statistically minded: Does limiting the range of results actually reduce the standard deviation?


Lakesidefantasy wrote:
I have a question for the statistically minded: Does limiting the range of results actually reduce the standard deviation?

Usually. It depends on the shape of the distribution as well.


The method I have players use is 4d6, drop lowest, reroll 1 & 2.

I'd be curious to see how that pans out. Generally leads to high stats for my players.

Edit: come to think of it, using (3d6, drop lowest) +6 would be the same thing, wouldn't it? Probably easier to enter into a rolling algorithm.


Scythia wrote:
Edit: come to think of it, using (3d6, drop lowest) +6 would be the same thing, wouldn't it? Probably easier to enter into a rolling algorithm.

I think 4d4 drop the lowest and add 6 would work to simulate your method. Decreasing the number of dice from 3 to 2 may broaden the curve.


Scythia wrote:

The method I have players use is 4d6, drop lowest, reroll 1 & 2.

I'd be curious to see how that pans out. Generally leads to high stats for my players.

Edit: come to think of it, using (3d6, drop lowest) +6 would be the same thing, wouldn't it? Probably easier to enter into a rolling algorithm.

I ran 10,000 simulations of your 3d6 drop the lowest and add 6 method. The average point-buy equivalent was around 42 points. The range was about 76 points, with the lowest of 10,000 being an 8 point-buy and the highest an 84 point-buy. The spread was a about a 12 point standard deviation.

By comparison, the Classic 3d6 method has a standard deviation of about 11 points when you don't eliminate the scores below 7.

I also ran 10,000 simulations of 4d4 drop the lowest and add 6 method. The average was 43 points, the range was 73 points, and the standard deviation was about 10 points.

Finally, 10,000 simulations of 5d3 drop the lowest and add 6 resulted in an average of 45 points, a range of 70 points, and a spread of around 8 points.

There appears to be an inverse relationship between average point-buy equivalency and standard deviation. As I bring the standard deviation down by adding more dice, I also bring the average up; which is bad for my goals.

I would like to see a random method that produces results with an average point-buy equivalent of around 20 points and a spread of about 5 points.


if you add xdn the mean should increase by x *((n+1)/2) (im not talking about pointbuy here just raw scores).
try to run 6d3 discard arrays with 6. mean should be around a 12 point buy.


Could one of you math savvy fellows tell me what the equivalent point buy would be for rolling 3d6, re-rolling anything less than 7, and replacing your lowest roll with a 16?

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
Lakesidefantasy wrote:
There appears to be an inverse relationship between average point-buy equivalency and standard deviation. As I bring the standard deviation down by adding more dice, I also bring the average up; which is bad for my goals.

You're confusing correlation with causation.

As you increase the number of dice, the standard deviation goes down.

The fact that the average score/point buy/whatever increases in the examples you have chosen is because that what happens when you choose "best n dice out of m rolls" - that's exactly what 'best' does.

If you want somewhere around the same average as 3d6, but with less standard deviation, you want to pick something rolling more than three dice, but with the average roll closer to the 10.5 expected from 3d6. Try 5d4 - 2.

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
thenobledrake wrote:
Could one of you math savvy fellows tell me what the equivalent point buy would be for rolling 3d6, re-rolling anything less than 7, and replacing your lowest roll with a 16?

There is no simple "equivalent point buy".

It's fairly straightforward to work out what the average point buy would be for "3d6, re-roll anything less than 7", but it's a fairly meaningless calculation. And you're never going to come up with a point buy system that can produce both a "7, 7, 7" and an "18, 18, 18" (or should that be "18, 18, 16"?)

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
Lakesidefantasy wrote:

I ran a top three of 10d6 for 10,000 trials and got a point-buy equivalent standard deviation of around 11 points, a median of around 65 points, and a range of around 80 points.

That is worse than "top three" of 3d6 with a standard deviation of around 9 points, median of around 3 points, and range of around 100 points.

It's only 'worse' than 3d6 because you're looking at it through the magnifying glass of a point buy scale that makes the difference between 17 and 18 appear to be larger than the difference between 11 and 12.

If you actually look at the raw ability scores you'll find that the standard deviation for "best 3 of 10" is lower than that for "best 3 of 3"


Sorry, that's what I meant to ask for was the point-buy equivalent of the average result of that particular rolling method.

I am wanting to see how that method compares to other rolling methods because it is one of a few that my group of players said they would be okay with using instead of using point buy.

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
thenobledrake wrote:
Sorry, that's what I meant to ask for was the point-buy equivalent of the average result of that particular rolling method.

That's fairly straightforward.

Thie distribution of results from rolling 3d6 is

  • 3: 1
  • 4: 3
  • 5: 6
  • 6: 10
  • 7: 15
  • 8: 21
  • 9: 25
  • 10:27
  • 11: 27
  • 12: 25
  • 13: 21
  • 14: 15
  • 15: 10
  • 16: 6
  • 17: 3
  • 18: 1

The average is 2268/216 = 10.5

If we discard scores below 7, that removes 1+3+6+10 results.
These contribute (1*3 + 3*4 + 6*5 + 10*6), or 105, to the total.

That makes the new average 2163/196, or just a shade over 11.

To get a score of 11 in 6 attributes would cost you 6 points.

I wouldn't suggest a 6-point buy, though; because point buy is based around 10, and rolling averages 10.5, I'd increase the 'equivalent' point total to perhaps 10.


JohnF wrote:
Lakesidefantasy wrote:

I ran a top three of 10d6 for 10,000 trials and got a point-buy equivalent standard deviation of around 11 points, a median of around 65 points, and a range of around 80 points.

That is worse than "top three" of 3d6 with a standard deviation of around 9 points, median of around 3 points, and range of around 100 points.

It's only 'worse' than 3d6 because you're looking at it through the magnifying glass of a point buy scale that makes the difference between 17 and 18 appear to be larger than the difference between 11 and 12.

If you actually look at the raw ability scores you'll find that the standard deviation for "best 3 of 10" is lower than that for "best 3 of 3"

Thanks JohnF, that makes sense. I ran the simulation again comparing the sum of the ability scores for each method. the "best 3 of 3d6" had a standard deviation of about 7 ability score points, the "best 3 of 10d6" was about 4 score points, and the "best 3 of 20d6" was about 2 points.

Back to point-but equivalency. The standard deviation for the "best 3 of 3d6" is around 11 point-buy points, for the "best 3 of 10d6" it is also around 11, and for the "best 3 of 20d6" it is about 7 point-buy points.

The standard deviation of 9 point-buy points for the "best 3 of 3d6" I quoted above was from Murph's analysis in the original post. Murph came up with a standard deviation of 9 and I came up with 11. This is because in my simulations I have not been eliminating ability score arrays that have scores less than 7.

Applying a lower limit like that appears to decrease the standard deviation, but I still want the possibility of rolling a 3-6 for an ability score. That makes it difficult to compare various methods of rolling scores against the point-buy method.

So, for calculating the point-buy equivalency in my simulations I extend the pattern of points for buying scores all the way down past 7 to 3. Here is the pattern I see, and this is the table of point buys I use in my simulations:

3 = -16
4 = -12
5 = -9
6 = -6
7 = -4
8 = -2
9 = -1
10 = 0
11 = 1
12 = 2
13 = 3
14 = 5
15 = 7
16 = 10
17 = 13
18 = 17


JohnF wrote:
Lakesidefantasy wrote:
There appears to be an inverse relationship between average point-buy equivalency and standard deviation. As I bring the standard deviation down by adding more dice, I also bring the average up; which is bad for my goals.

You're confusing correlation with causation.

As you increase the number of dice, the standard deviation goes down.

The fact that the average score/point buy/whatever increases in the examples you have chosen is because that what happens when you choose "best n dice out of m rolls" - that's exactly what 'best' does.

If you want somewhere around the same average as 3d6, but with less standard deviation, you want to pick something rolling more than three dice, but with the average roll closer to the 10.5 expected from 3d6. Try 5d4 - 2.

I expected more dice to bring the spread of the results down. That's why I was surprised at my earlier results. I thought it may have something to do with rolling more dice but only adding up the result of the same number of dice (3) each time. However, you are right, it was the peculiarities of the point-buy method that was masking the standard deviation decreasing effect of rolling more dice.

So, 10,000 simulations of rolling 5d4-2 to generate ability scores results in a point-buy equivalency standard deviation of about 9 points, with an average of of around 3 points, and a range of about 79 points.

In comparison, 10,000 simulations of the Classic 3d6 method results in a point-buy equivalency spread of around 11 points, an average of about 3 points, and a range of 104 points.

You can see that 5d4-2 has a lower standard deviation of 9 points.

Note: An average point-buy result of 3 is to be expected of a method that does not have a bias in the results. For instance, an ability score array of (10, 10, 10, 11, 11, 11) would equal 3 points.

Anyway, a spread of 9 points is still higher than I would like. My ultimate goal is to find a random rolling method that would be competitive with a 15 or 20 point-buy while still allowing for the full range of results between 3 and 18 for any particular ability score.

It seems to me that we can try and manipulate several of the characteristics of the results for any particular method by adding number of dice rolled for decreasing the standard deviation, or taking the "best of x dice" to increase the overall average. Other characteristics that may be manipulable could be the range of a particular method. For instance, eliminating ability score arrays that add up to less than 10 points would limit the range in a biased manner.


Adamantine Dragon wrote:

I prefer the James Tiberius Kirk approach to stat generation:

"Never tell me the odds!"

Isn't that Han Solo? Or was there a Trek episode I missed somewhere?

As for the topic, thank you to the more math minded folks out there. This has been extremely illuminating.


1 person marked this as a favorite.

I've always liked 1d8+8 (although I most prefer PB). Nothing less than a 9, but nothing more than a 16 (before racials).
What does 1d8+8 give us Murph?


The trouble with comparing point buy and rolling is that at the high and low ends they don't follow the same rules.

The average stat from 4d6 drop low is around 12.2 I think. That is the same as about a 13 point buy... BUT only at the average values, The dice don't typically give you all 12s, and pb costs for very high or very low score wildly affect the comparison. As was pointed out to me by an avid point buyer: "With point buy your main stats can be as high as you wish, while when you roll you often don't get stats as high as you might like."

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber

It shouldn't be necessary to use Monte Carlo simulation methods to approximate the statistics of most of the dice rolling methods being suggested here; the problems are small enough that the exact value can be calculated.

For "Best 3 of 4d6", one can just run through the 6*6*6*6=1296 possible cases. But for "Best 3 of 10d6" that's 60,466,176 ways of rolling the dice, which is no longer trivial. For 20d6 the number goes up to over three million billion; the NSA could probably handle that amount of crunching, but it's beyond the capabilities of the average home computer.

Fortunately, there are ways to massively reduce the amount of calculation needed.

(In the following examples, I'm only going to talk about "Best 3 of n d6", but the principles involved can be readily extended to other cases)

The first observation is that we don't really even need to look at all 216 (6*6*6) possible ways of rolling 3d6; if we have a 3, a 4, and a 5, say, we don't really care about the order in which they were rolled. It turns out that there are only 56 distinct outcomes for 3d6 if we ignore the order in which the rolls were made.

Secondly, we can get from "Best 3 of n d6" to "Best 3 of n+1 d6" by simply looking at each of those 56 outcomes, and seeing what happens when we roll another d6. Is the new value large enough to displace one of the previous 3 best? If it is not, then the "best" remains unchanged; if it is, then we have attained a new best outcome.

That means we only have to look at 56*6 cases, repeated 17 times, to go from "Best 3 of 3d6" to "best 3 of 20d6" (roughly a million-million-fold reduction in the amount of calculation to be done).

For anyone who wants them, the raw distributions of scores for best 3 of 4/10/20 d6 are:

4d6 10d6 20d6
3 1 1 1
4 4 10 20
5 10 55 210
6 21 1068 1048765
7 38 5165 10485950
8 62 16685 60293310
9 91 69580 3535543355
10 122 208295 23295036490
11 148 498430 96894443660
12 167 1268087 1169574442975
13 172 2708290 5547872994900
14 160 5066920 18457353788920
15 131 8737964 101753402753539
16 94 13665445 362841857950870
17 54 14629005 711735779888910
18 21 13591176 2454528801391101

Silver Crusade

Rolling dice for stats can lead to extreme variation, but that's not always bad. I'm playing in an ongoing 3.5 home game where we rolled dice (4d6, use best 3 dice, for each attribute. No re-rolls).

My friend rolled 18, 18, 16, 16, 16, 12. That's a 50+ point buy.

I rolled 14, 12, 12, 12, 12, 9. That's a 12 point buy.

Despite the extreme disparity in attributes, the characters work together just fine. I built an summoning reach cleric, which is a very strong build. My friend built a fighter. It's probably best I started with low attributes, that way I don't overshadow the others and don't need to hold back.

Shadow Lodge

Pathfinder Lost Omens, Maps, Rulebook Subscriber
Lakesidefantasy wrote:
Adding dice does not seem to reduce the spread of the results.

The number of dice rolled doesn't bring the spread of the results down.

If you're keeping three six-sided dice, then no matter how many dice you roll, and how you select the three dice you're going to keep, the spread of results will range from three '1's to three '6's. All you're doing by varying the number of dice rolled or the way you choose the three dice to keep is changing the relative probabilities of each of the results.


JohnF wrote:
Lakesidefantasy wrote:
Adding dice does not seem to reduce the spread of the results.

The number of dice rolled doesn't bring the spread of the results down.

If you're keeping three six-sided dice, then no matter how many dice you roll, and how you select the three dice you're going to keep, the spread of results will range from three '1's to three '6's. All you're doing by varying the number of dice rolled or the way you choose the three dice to keep is changing the relative probabilities of each of the results.

I think I understand. Does that mean that 5d4-2 is the only other option we have? I tried to think of other rolling methods that generate results from 3 to 18, but it appears that only 3 exist: 1d16+2, 3d6 and 5d4-2.


Aranna wrote:
"With point buy your main stats can be as high as you wish, while when you roll you often don't get stats as high as you might like."

That is what is good about point buy. What I am curious about is the possibility that there may be a rolling method that does "often" give you stats as high as you might like. That's why I'm interested in a method that has a very narrow distribution curve.

There are 54,264 possible ability score arrays, but with something like a 20 point buy there are only 280. If you don't dump any of your scores you're down to 46 arrays.

What I would like to know, is how low would a point buy value need to go before something like 5d4-2 looks better?

In the end I would like to see a hybrid method that gives you the control to get what you want, but also gives you the possibility of getting even better scores. Maybe some sort of risk mitigation method?


Lakesidefantasy wrote:
JohnF wrote:
Lakesidefantasy wrote:
Adding dice does not seem to reduce the spread of the results.

The number of dice rolled doesn't bring the spread of the results down.

If you're keeping three six-sided dice, then no matter how many dice you roll, and how you select the three dice you're going to keep, the spread of results will range from three '1's to three '6's. All you're doing by varying the number of dice rolled or the way you choose the three dice to keep is changing the relative probabilities of each of the results.

I think I understand. Does that mean that 5d4-2 is the only other option we have? I tried to think of other rolling methods that generate results from 3 to 18, but it appears that only 3 exist: 1d16+2, 3d6 and 5d4-2.

One more, 15d2-12.

10,000 ability scores arrays generated with this method results in a point buy equivalency standard deviation of about 6, and a range of only 55 points.

Histogram of 15d2-12:

-26:*
-25:*
-24:*
-23:
-22:*
-21:*
-20:*
-19:
-18:*
-17:*
-16:*
-15:*
-14:*
-13:*
-12:*
-11:**
-10:***
-9:****
-8:******
-7:*******
-6:*********
-5:*************
-4:****************
-3:********************
-2:***********************
-1:**************************
0:*****************************
1:*********************************
2:**************************************
3:*************************************
4:*************************************
5:*********************************
6:********************************
7:****************************
8:***********************
9:*********************
10:**************
11:***********
12:*********
13:*******
14:*****
15:***
16:***
17:**
18:**
19:*
20:*
21:*
22:*
23:
24:*
25:
26:
27:
28:*
29:*

Although it is a bit absurd to roll d2s, the lions share of the resulting ability score arrays fall within a 12 point spread. That's the difference between a 15 and a 27 point buy. Not so bad.

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