Story
2014-07-16, 01:08 AM
I've seen statements before about the average stats when rolled randomly, such as the assertion that it corresponds "almost exactly to 28pb". However, these are wrong, because they overlook the rules for rerolling low stats.
The PHB says to reroll if the highest stat is 13 or less or the total bonus is 0 or less. This has about a 12.67% chance, so it noticeably affects the stats.
Without rerolling, the average PB is 28.53 (1027/36), which still isn't "almost exactly" 28, but I can see how people got the idea. But with rerolling, the average PB is 30.46 (precisely 63026644128836649177/2068935363829252540). This is skewed upwards a bit, so the median is just over 29.
Incidentally, the average value of the nth highest stat are
15.9487203989
14.4547736243
13.2274776521
12.0230480882
10.6575440448
8.72031395942
So if you want an elite array that represents the average rolls, 16, 14, 13, 12, 11, 9 is much more reasonable (11 and 9 are rounded up to fit the median PB of 29)
The distribution of highest score is
14: 0.129484657409
15: 0.23618891612
16: 0.296377639957
17: 0.232018943187
18: 0.105929843326
So natural 18s aren't unheard of, but you can't rely on them either. A rule of thumb is that 14/15, 16, and 17/18 are about equally likely.
The odds of ending up with a 3 are around 1 in 2.7 million. 3 is already the least likely occurrence (just 1 in 1296 for any given stat) but the rerolling rules make them significantly less likely since any stat array containing a 3 will almost certainly be rerolled. You're more likely to have a 17 as your lowest stat than a 3.
A bit of trivia: There are 25,204 possible combinations of stat values. The least likely is 18,18,18,4,3,3, which has a 1 in 1.8 trillion chance of occurring. This is about 40 times less likely than getting all 18s.
The PHB says to reroll if the highest stat is 13 or less or the total bonus is 0 or less. This has about a 12.67% chance, so it noticeably affects the stats.
Without rerolling, the average PB is 28.53 (1027/36), which still isn't "almost exactly" 28, but I can see how people got the idea. But with rerolling, the average PB is 30.46 (precisely 63026644128836649177/2068935363829252540). This is skewed upwards a bit, so the median is just over 29.
Incidentally, the average value of the nth highest stat are
15.9487203989
14.4547736243
13.2274776521
12.0230480882
10.6575440448
8.72031395942
So if you want an elite array that represents the average rolls, 16, 14, 13, 12, 11, 9 is much more reasonable (11 and 9 are rounded up to fit the median PB of 29)
The distribution of highest score is
14: 0.129484657409
15: 0.23618891612
16: 0.296377639957
17: 0.232018943187
18: 0.105929843326
So natural 18s aren't unheard of, but you can't rely on them either. A rule of thumb is that 14/15, 16, and 17/18 are about equally likely.
The odds of ending up with a 3 are around 1 in 2.7 million. 3 is already the least likely occurrence (just 1 in 1296 for any given stat) but the rerolling rules make them significantly less likely since any stat array containing a 3 will almost certainly be rerolled. You're more likely to have a 17 as your lowest stat than a 3.
A bit of trivia: There are 25,204 possible combinations of stat values. The least likely is 18,18,18,4,3,3, which has a 1 in 1.8 trillion chance of occurring. This is about 40 times less likely than getting all 18s.