Disclaimer: Before we go any further, I must state that despite looking at this topic, the wording, and the overall theme, my aim is not to get all DMs, GMs, referees, judges, and players to attain the highest possible ability scores for the characters. I am not going to side with low or high scores or any of the methods or play-style. As a roleplayer, I appreciate that the goal of the game is to have fun. How you achieve that, what means you need to employ to make that happen, is totally up to you and the people you are playing with.
If you are only here to find out if the Standard Array or the 4d6-drop-lowest method produces the highest values, then let me tell you the outcome so that you can be on your way. The winner is the 4d6-drop-lowest method, but by just a hair.
Now, if you are still interested in the occult sciences behind the scenes, then I think we can get started, and I can tell you what I will be examining here.
Some people tend to think that the expected value you can get on a normal six-sided-die (d6) in the long run is 3. That is close but not true. The lowest value on the die is a one, and the highest is a six. If you average that by doing the math [(1 + 6)/2], you get 3.5.
This logic is also what 5e is using, not just for the d6 but also for all other dice. You can see the same thing at work at many places, for example: where they offer a fixed hit point for the characters, offering 4hp for anyone getting d6, 5hp for anyone getting d8, 6hp for d10, and 7hp for d12. What they do is round up to the nearest value. You can also see this in monster stats, where they propose a hit point amount or a fixed value for damage. Both for hit point and damage, they round down these values, so a 3d8 hit point ends up being 13 (4.5 + 4.5 + 4.5 = 13.5 rounded down to 13), and a 1d6+2 damage is just 5 (3.5 + 2 = 5.5 rounded down to 5).
Simply looking at the above section, it would be easy to see that the expected value of 3d6 is 10.5, which is just the sum of the three individual dice: 3.5 + 3.5 + 3.5.
Using 3d6 and using all three dice, you can achieve ability scores in the range of 3-18. Not all values in this range have the same chance to show up when you are creating a character, however. Now, why could that be?
Well, as we discussed above, on a d6, you can have six different values or combinations: 1, 2, 3, 4, 5, and 6. What happens if you add another d6 to this? To make it easier, let’s color the dice. We’ll have the red d6 and the green d6, and let’s also say that we want to see all the different combinations possible with these two dice. I will immediately tell you that I can produce 36 different combinations. How this would go is, I’ll set the red d6 to 1 and the green d6 to 1 as well at the start. Then I’ll turn the green d6 to 2, then 3, 4, 5, and 6. I will go ahead and turn the red d6, which so far was resting peacefully on 1, to 2 and turn the green d6 back start, which was a 1. I’ll continue turning the green d6 to 2, 3, 4, 5, and 6, and when the cycle is completed, I will again adjust the value of the red die. The above process is 6 x 6 possible combinations, and it covers all possibilities. If you add the third d6 (blue) (and don’t worry, I will not start with the above turning cycle again), the variations will increase to 6 x 6 x 6, which is 216. So, we finally reached where I was heading – the 3d6 yields 216 different device combinations for 16 different values (3 – 18 range).
The trick here is that you can construct some, but not all, the different values in several ways. You can only get a 3 if you have all three dice showing ones (1,1,1), and you can only get an 18 if all dice are sixes (6,6,6). The odds are 1:215, which means that your chances are 0.463% – not too great, but don’t worry. Based on statistical analysis, if you roll enough 3d6s and look back on the values you got, every 216th roll will be a 3 or an 18. Keep rolling! The closer you get an infinite number of rolls, the more accurate this is going to be!
How to get the percentage? Divide the number of combinations by the total number of combinations and multiply the end by 100 for percentages - so for a 3 or 18, it is 1/216*100.
So how about a 4? That’s tricky. You could get a 4 by having either the red, green, or blue die showing a 2, all the rest a 1 (1,1,2, or 1,2,1, or 2,1,1). There are 3 combinations out of the 216 that could give you a value of 4. The odds of getting a value of 4 are 3 to 213, which is a 1.39% chance.
The growing number of combinations nearing the expected value of 10-11 also raises the chances that you get more values around 10-11, of course. When to put graphic format, this produces the famous bell curve (below).
The below table shows the chance of rolling a given value using 3d6.
Introducing the 4th. Adding a fourth d6 into the mix raises the possible combinations to 6 x 6 x 6 x 6, 1296. Remember, however, that you are still constructing values in the range of 3 to 18; we are just removing the lowest value die out of the 4 and using the 3 highest. I want to point out two things right at the start:
4d6-drop-lowest is a cunning little scoundrel that starts at the same place as the standard 3d6 but leaps much higher. And we haven’t even looked at what’s in the middle!
This method distorts the bell curve and raises the expected value of the rolls from 10.5 to a whooping 12.2446. So while using 3d6, the most ability scores would be around 10 and 11; switching to 4d6-drop-lowest, these would raise to 12 and 13.
The below table shows the different combinations and chances.
This table shows 3d6 and 4d6-drop-lowest on the same graphic so that you can compare the distortion. The orange/amber line is the standard 3d6, while the blue represents the 4d6-drop-lowest version.
The standard array is nothing but the ubiquitous permutation of the point-buy system in 5e, where you have 27 points to customize your ability scores using those. The actual values are 15, 14, 13, 12, 10, 8, yielding an average of 12. 12 is just slightly worse than the 4d6-drop-lowest method that we discussed above. If you want to close the gap between the two versions, add one or two points to the standard array. This seems to be a hot topic on different forums, so purely from a mathematical point of view, I would suggest this. Adding one would make the average of the standard array 12,167 while adding two would mean 12,333. Neither is a perfect fit, of course.
I am sure there is a lot more to say about combinations, averages, percentages, and odds. I believe this article was already much more than a sane role-player would want to know about the wheels of science slowly turning in the engine of the 5th edition game. Still, I hope I could help out the curious and made no fatal mistake in my calculations!