There are many ways to generate your character’s ability scores. The PHB provides 3 ways to do this: rolling, point buy, and the standard array. I’m going to talk about the pros and cons of each method and give a few more ways you can generate your ability scores.
Probably the most popular method and one of the two recommended ways generating ability scores, rolling is the most controversial method of the bunch. When each player rolls their ability scores, all characters are going to have different power levels. You might have a player with several 16+ scores and another whose highest score is 12. You might have a player who has all negative modifiers and another player who has all positive. If you decide to roll ability scores, you must accept this.
You must also accept that if you are the DM and you allow your players to roll their scores away from your eyes, you will have at least 1 player that pads their numbers, maybe adding a 1 to every result or changing one of their rolls to a 17 or 18 to ensure they get the maximum benefit of the rolling method. This is where the controversy occurs. It is usually obvious when a player has falsified their rolls, and allowing such a character can have a negative impact on everyone else. If you are starting the game at level 1 and you have a player who already has max stats in their primary attributes, then they will be single-handedly destroying most CR-appropriate combat encounters thrown at the party, taking the fun away from everyone else including the DM.
Point buy is the second most popular method, just behind rolling. It is popular because it assures the players and DM that all of the players will be starting off at about the same power level. While there will be certain encounters that cater to certain classes or playstyles, no one character should really constantly be in the spotlight for too long.
Point buy does allow players to allocate their attributes the way they want to, so the randomness of D&D is reduced at a cost of more balanced fun for all. The CR system will be more accurate as well so long as the party isn’t stockpiled with powerful magic items or other such boons.
The standard array represents what an average adventurer consists of. It will give all players about 2 good stats, 2 average stats, and 2 poor stats. You can arrive at the standard array by using the point buy method as well. Pretty much the same things said about point buy can be said about the standard array, except that the standard array is a bit more limiting. If you want to have, say, two 15’s, or no negative modifiers, you can’t whereas with the point buy system, you could accomplish this.
This is a method I’ve used several times, and I have gotten mostly positive feedback from it. I generally use this method for games that do not start at level 1. Essentially, I create an array of values for my players and they get to assign each value to an attribute. Some examples of DM arrays would be 16, 16, 12, 12, 10, 10 or 17, 15, 13, 11, 10, 8. The DM array can be generated by rolling or by picking and choosing what numbers you want to allow your players to have access to at level or whatever level you end up starting your game at.
This is another method I’ve used a few times, and it sort of combines the rolling and standard array methods. What you do is you roll 6 sets of ability scores and arrange them in a 6×6 grid in the order you rolled them. Then, you pick a row or column from the grid and use those values for your ability scores. An example grid would be as follows:
The following would be a few example sets of ability scores you could use from the above matrix:
9, 17, 16, 10, 7, 15 (1st row)
13, 13, 16, 10, 8, 12 (3rd row)
17, 14, 13, 16, 14, 14 (2nd col)
16, 18, 16, 4, 13, 18 (4th col, and ironically what your average rolled character looks like)
Rolling Scores In Order
I have never tried this method as I always have a character concept in mind before I start determining ability scores. Rolling scores in order uses the rolling method but instead of rolling all 6 values and then determining which score should go to which ability, the scores are assigned in the order they were rolled starting with Strength, then Dexterity, Constitution, Intelligence, Wisdom, and lastly Charisma. An added stipulation with this method is you can only roll once, so there is no chance to try rolling until you get (close to) what you want.
Using the matrix above, let’s say you rolled the first row: 9, 17, 16, 10, 7, and 15. 9 would be assigned to Strength, 17 would be Dexterity, 16 would be Constitution, 10 would be Intelligence, 7 would be Wisdom, and 15 would be Charisma.