Whenever you create a D&D character, you use die rolls to determine your characters stats. I was curious about how the stats are distributed and the probability of obtaining particular results. To investigate this, I wrote two programs, one iterated through all of the possible rolls for 4 die, dropped the lowest, and found the stat total. This let me figure out the mean, mode, and quartiles for the distribution of the stats.. In short, out of all the possible rolls, the number of outcomes for each number is as follows:
- 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
- Mean: 12.24
- Modifier Mean: 0.87
The most common roll is a 13, although 12 and 14 are also very common. The lowest 25% of the rolls are 10 or less. The second lowest 25% are between 10 and 12. The next highest 25% is between 12 and 14. The highest 25% are between 14 and 18. The modifier mean is the average ability score modifier you will get from any particular roll. The most interesting result for me was the modifier mean. I expected it to be higher. If you've ever wondered why characters made under the standard system seem underpowered, that might be why.
BUT WAIT, THERE'S MORE! Let's say that you roll your 6 stat scores. If you ordered the stats from lowest to highest, the probability of each score having a particular value is different. It's much more likely that your low score will be a 6 than for your high score to be a 6. I created a table that lays out the probability that each score will be a particular number. This table was generated by rolling 1,000,000 sets of of ability scores and tallying the results. I have also listed the mean and median for the scores and the average stat modifier for each.
Roll 4 Drop 1
| % of rolls | Lowest | 2nd | 3rd | 4th | 5th | Highest |
| 3 | 0.46 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4 | 1.84 | 0.02 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 | 4.47 | 0.17 | 0.00 | 0.00 | 0.00 | 0.00 |
| 6 | 8.78 | 0.87 | 0.04 | 0.00 | 0.00 | 0.00 |
| 7 | 14.17 | 3.14 | 0.29 | 0.01 | 0.00 | 0.00 |
| 8 | 18.81 | 8.20 | 1.49 | 0.14 | 0.01 | 0.00 |
| 9 | 19.97 | 15.94 | 5.22 | 0.90 | 0.08 | 0.00 |
| 10 | 16.30 | 22.76 | 13.05 | 3.80 | 0.57 | 0.03 |
| 11 | 9.71 | 22.93 | 22.09 | 10.87 | 2.71 | 0.28 |
| 12 | 4.13 | 16.15 | 19.79 | 21.10 | 8.79 | 1.50 |
| 13 | 1.16 | 7.39 | 19.79 | 27.02 | 18.83 | 5.43 |
| 14 | 0.18 | 2.07 | 9.39 | 22.02 | 26.86 | 13.41 |
| 15 | 0.01 | 0.30 | 2.52 | 10.89 | 24.31 | 22.64 |
| 16 | 0.00 | 0.02 | 0.32 | 2.89 | 13.54 | 26.68 |
| 17 | 0.00 | 0.00 | 0.02 | 0.35 | 3.91 | 20.62 |
| 18 | 0.00 | 0.00 | 0.00 | 0.01 | 0.39 | 9.39 |
| Mean | 8.50 | 10.41 | 11.76 | 12.95 | 14.17 | 15.66 |
| Median | 9 | 10 | 12 | 14 | 14 | 16 |
| Modifier Mean | -1.00 | -0.04 | 0.63 | 1.23 | 1.84 | 2.59 |
In case you don't know, the median is the halfway point in the data. So for example, there is a 50% chance that your highest score will be 16 or less, and a 50% chance that it will be 16 or more. Even though the average modifier for any roll is 0.87, the average modifier for your highest roll is 2.59. This means that even though your ability scores might not be impressive overall, your highest stat should still be good.
I repeated the experiment, but this time I used the the more powerful roll 5 drop 2 method of making character scores.
I repeated the experiment, but this time I used the the more powerful roll 5 drop 2 method of making character scores.
Roll 5 drop 2
| % of rolls | Lowest | 2nd | 3rd | 4th | 5th | Highest |
| 3 | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 4 | 0.38 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 5 | 1.13 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 6 | 3.10 | 0.08 | 0.00 | 0.00 | 0.00 | 0.00 |
| 7 | 6.50 | 0.40 | 0.01 | 0.00 | 0.00 | 0.00 |
| 8 | 11.24 | 1.76 | 0.12 | 0.00 | 0.00 | 0.00 |
| 9 | 16.73 | 5.32 | 0.72 | 0.05 | 0.00 | 0.00 |
| 10 | 20.41 | 12.40 | 3.10 | 0.40 | 0.03 | 0.00 |
| 11 | 18.86 | 20.70 | 9.34 | 2.15 | 0.25 | 0.01 |
| 12 | 13.19 | 25.24 | 19.74 | 7.96 | 1.63 | 0.14 |
| 13 | 6.23 | 20.43 | 27.73 | 19.21 | 6.80 | 0.99 |
| 14 | 1.83 | 10.42 | 24.36 | 29.43 | 18.44 | 4.70 |
| 15 | 0.28 | 2.81 | 11.86 | 26.24 | 30.12 | 14.38 |
| 16 | 0.01 | 0.33 | 2.79 | 12.26 | 28.45 | 28.25 |
| 17 | 0.00 | 0.01 | 0.21 | 2.19 | 12.54 | 32.02 |
| 18 | 0.00 | 0.00 | 0.00 | 0.08 | 1.72 | 19.49 |
| Mean | 9.93 | 11.79 | 13.06 | 14.14 | 15.20 | 16.43 |
| Median | 10 | 12 | 13 | 14 | 15 | 17 |
| Modifier Mean | -0.28 | 0.65 | 1.28 | 1.82 | 2.35 | 2.98 |
We all know that the roll 5 drop 2 method produces better ability scores, and this data gives us some numbers to compare using. Look at the probabilities for your highest score. There is a 79.76% chance of rolling a 16, 17 or 18. That's amazing.
Dungeon Masters, hopefully this gives you some insight into different ways that you can allow your players to roll their ability scores. Are their any other methods that you would like for me to test in another post? Let me know in the comments and I'll simulate them for a later post.
Dungeon Masters, hopefully this gives you some insight into different ways that you can allow your players to roll their ability scores. Are their any other methods that you would like for me to test in another post? Let me know in the comments and I'll simulate them for a later post.
