Ability score statistics: point buy vs. rolling

[kerbarian]

After reading the recent thread on point buy, I was curious about how point buy compares to dice rolling in terms of overall stats, so I wrote a program to do some calculations.

If you roll 4d6 and drop the lowest 6 times and then work out how much those stats would have cost you in the point buy system [1], it comes out to 25.1 points. However, the rules let you reroll if your stats are too low. Taking that into account (which was much more complicated), the average cost becomes 26.3 points. As a side note, the chance of having stats low enough to allow a reroll is 7.9%.

Since you get more control, I'd say that the standard point buy is pretty comparable, on average, to the standard dice rolling. That's assuming your DM follows the rules strictly, though, and makes you keep 14, 10, 10, 10, 10, 8 if you roll it. Most DMs I know are somewhat more lenient with rerolls, which would shift things in favor of dice rolling.

I also worked out the average points value for the "High-Powered Characters" method -- 5d6 and drop the lowest 2. The average point buy cost for the stats you'd get with that method is 38.3 (including rerolls, which are only 1.4% even with the higher thresholds). That's much higher than the "High-powered campaign" points value of 32 for point buy.

So if your DM ever gives you a choice between the DMG versions of high-powered point buy or high-powered dice rolling, go with the dice.

With the programs I just wrote, I also have the ability to come up with all kinds of other statistics about dice rolling, if anyone has any numbers they've been curious about.


[1] Rolling dice lets you get scores less than 8, which you can't get with the point buy system. To make the comparison, I gave the scores 3 through 7 points values of -5, -4, -3, -2, and -1.

 

[frankthedm]

Your forgeting players making choices that get them killed to get another chance to roll scores.

I prefer to give PCs; 17, 16, 15, 14, 13, 12; lower as desired.

 

[Geoff Watson]

The average 'point buy' value for '4d6 take best 3' ability scores is between 29 and 30, depending on how you handle low scores and rerolling really bad scores.

You can't just take the value of the median score, as the values aren't symettrical.

Geoff.

 

[kerbarian]

The average 'point buy' value for '4d6 take best 3' ability scores is between 29 and 30, depending on how you handle low scores and rerolling really bad scores.

You can't just take the value of the median score, as the values aren't symettrical.

I definitely didn't just do that. For the simple version (4d6, drop the lowest, not considering rerolls due to low stats), the probabilities for rolling each ability score work out to be:

Score   %Chance  Cost
    3   0.0772%    -5
    4   0.3086%    -4
    5   1.1574%    -3
    6   2.3148%    -2
    7   3.8580%    -1
    8   6.1728%     0
    9   8.7191%     1
   10  10.4938%     2
   11  12.1914%     3
   12  12.7315%     4
   13  12.4228%     5
   14  10.7253%     6
   15   8.4877%     8
   16   5.7870%    10
   17   3.3179%    13
   18   1.2346%    16

If you multiply the probability of each stat by its corresponding cost and add them up, you get the average point cost for a 4d6-drop-the-lowest roll, which is 4.1875. Multiply that by 6 stats, and the average point cost for an entire set of standard-rolled stats is 25.1250.

To account for rerolls due to low stats, I calculated all possible sets of stats and their corresponding probabilities. For each set, I also worked out its total point buy cost and whether or not it could be rerolled (max stat <= 13 or sum of modifiers <= 0). If you ignore all sets of stats that can be rerolled, the remaining average point buy cost for each possible set of stats, weighted by the probability of rolling that set, is 26.3121.

 

[KarinsDad]

Sorry, but your numbers are off slightly.

For example, there are 21 different ways to roll an 18 out of 1296 different possible rolls. This is easy to prove (e.g. 1 of 6 6 6 6, 5 of 1-5 6 6 6, 5 of 6 1-5 6 6, 5 of 6 6 1-5 6, 5 of 6 6 6 1-5).

So, that's a 1.62% chance of rolling an 18, not a 1.23% chance as per your table. Your table is obviously incorrect.

The actual point buy for your system (i.e. assigning a -1 for a 7, etc.) results in a 28.52778 point buy. If you assign 0s for 3s through 7s, it comes out 29.13426.

This does not take into account re-rolls where the point buy increases by several points.


Btw, the median score for 4D6 drop lowest is 12.2446.

Also, 5D6 drop lowest two works out to 38.04167 average point buy with a median of 13.43017.


To make sure that my answer was correct, I had several different cross checks in my excel spreadsheet. For example, after finding out how many 3s, 4s, ..., and 18s came up, I added them up and it came to 1296. I even did the final calculations two different ways to make sure I came up with the same answer. I have a high confidence that my answer is accurate, although there is still a chance that I have an error somewhere as well since I did not actually write a program where I could check the code, instead I am using a spreadsheet (where if the data in a cell is inaccurate, it could change the answer).

 

[(Psi)SeveredHead]

So if your DM ever gives you a choice between the DMG versions of high-powered point buy or high-powered dice rolling, go with the dice.

I'm not all that interested in high stats, and having more control over character design is a plus for me.

It's been my experience that dice-rolling usually gets you higher stats, probably due to generous die-rolling (drop 1s, for instance), but of course someone nearly always gets screwed.

 

[kerbarian]

Sorry, but your numbers are off slightly.

For example, there are 21 different ways to roll an 18 out of 1296 different possible rolls. This is easy to prove (e.g. 1 of 6 6 6 6, 5 of 1-5 6 6 6, 5 of 6 1-5 6 6, 5 of 6 6 1-5 6, 5 of 6 6 6 1-5).

So, that's a 1.62% chance of rolling an 18, not a 1.23% chance as per your table. Your table is obviously incorrect.

The actual point buy for your system (i.e. assigning a -1 for a 7, etc.) results in a 28.52778 point buy. If you assign 0s for 3s through 7s, it comes out 29.13426.

This does not take into account re-rolls where the point buy increases by several points.

Ah, you're right. The algorithm was correct, but I had a typo as to which dice to add up in one case. Specifically, if the 4th die was lowest, it was adding up dice 2,3,4 instead of 1,2,3. With that fixed, I also get 28.5278 for the average. When accounting for rerolls (which will happen 3.8114% of the time, not 7.9%), it ups the average to 29.1854.

So I guess that changes my original point. Dice rolling will give you a noticably better average result than standard point buy.

I also just compared "Low-powered" point buy to straight 3d6 rolling. Straight 3d6, 6 times gives an average point buy cost of 16.1111. If you account for rerolls (5.478% chance of max stat <12 or modifier sum <= -3), it ups the average to 16.8528.

Also, 5D6 drop lowest two works out to 38.04167 average point buy with a median of 13.43017.

Yes, 38.0417 is also what I had for the 5d6-drop-2 average (no typo in that one). With the 1.3759% chance of a reroll, though, it ups the average to 38.3380.

For that, I followed the wording in the DMG on rerolls, which is a little weird. For standard rolling, you can reroll if "your highest score is 13 or lower." For high-powered rolling, you can reroll if your result "doesn't have at least one score of 15 or higher." Which means that a 14 disqualifies you for rerolls under standard and a 15 does for high-powered (assuming the sum of modifiers is high enough).

 

[Wolf72]

I prefer 4d6 drop lowest ... we do have a house rule though, reroll any ones ... 3 ones = 18, 4 6's or 4 1's = 19.

definitely a bit higher than normal, but still able to get a low score (in our last game my pc had a 9 dex and the rogue had a 7 cha)

 

[kerbarian]

I prefer 4d6 drop lowest ... we do have a house rule though, reroll any ones ... 3 ones = 18, 4 6's or 4 1's = 19.

definitely a bit higher than normal, but still able to get a low score (in our last game my pc had a 9 dex and the rogue had a 7 cha)

A number of people have mentioned the house rule of rerolling ones. I decided to see how much of a difference it makes, and it's pretty huge.

Rolling 4d6 and dropping the lowest gives you an average point buy cost of 28.5 (29.2 with rerolls). If you reroll ones, the average point buy cost jumps all the way to 37.4 (37.5 with rerolls). That's about as good as 5d6, drop the lowest 2.

 

[Asmor]

A bunch of people, myself included, tackled this a long time ago.

http://www.enworld.org/showthread.php?t=128481

Short, possibly biased, answer: The average rolled character is just under 30.5 points. This is assuming that a 7 is -1, a 6 -2, etc.

There was a bit of variation, but after checking both an "exact" method and the "monte carlo" method of rolling up a million characters and averaging them, both answers came out close enough that I'm pretty confident in the 30.5.