When Wizards of the Coast published the 3rd edition of Dungeons and Dragons (and thus set down the foundation for d20 games) they adopted a new, standardized, open-ended schedule of attribute modifiers (as compared to the fixed-range and sometimes attribute-specific schedules from earlier editions). In order to be open-ended the modifiers had to be defined by a formula – mathematically it’s

Floor(attribute/2) – 5

In plainer language, it’s +1 for every 2 points above 10, -1 for every 2 points below 11. Simple enough, but there’s a drawback: the “every 2 points” bit (necessary to keep attribute modifiers from becoming completely overwhelming differences between characters, overriding most other differences in ability) means that odd-valued attributes are basically useless – a 13 is worth exactly as much as a 12, a 5 is just as bad as a 4. These values seem like vestigial bits from prior editions, and while the designers of 3E tried to ameliorate that by giving them some utility (defining attribute requirements for feats in terms of odd numbers) they weren’t very successful – for example, substandard attribute values don’t end up differentiated, and feats a player wasn’t considering can’t make their character’s odd attribute values meaningful.

To fix this without upsetting the apple cart (all the other design work in 3E/d20), we’d really like to give these odd attribute values a modifier that’s halfway between that of the even values that bookend them, so for example, a 12 => +1 and a 14 => +2, so a 13 => +1.5. Fine, except what does that mean? How do you add half a point to a roll? You can say something about using it as a tie-breaker, but that won’t come up very often, and even when it does there’s no guarantee the opposition won’t have an odd attribute and the accompanying tie-breaking modifier on their end of things.

How about this: let’s get to that “.5” modifier statistically. That is, instead of adding half a point to a bunch of rolls, let’s add 1 point about half the time. We don’t even need to do any extra work for this – the rule could be

When your die roll is an odd number, add 1 to it for every odd-valued attribute that would modify the roll.

As an example let’s look at a Paladin making a Fortitude save. They have a Con of 13 (+1) and a Charisma of 15 (+2), and because they’re a Paladin their Charisma bonus applies to all their saves. When they roll a Fortitude save, they’ll add +3 when their roll is even, and +5 when the number they roll is odd. On a roll based on Charisma alone (Diplomacy, perhaps) the Paladin’s attribute bonus will be +2 half the time and +3 the other half, averaging out to +2.5. This makes their 15 Charisma worth exactly half the difference between a Charisma of 16 and a Charisma of 14, and doesn’t require us to rewrite the rest of the game.