Bell Curve Rolls
Here's perhaps the most fundamental variant to the d20 rules: Don't use a d20! Instead, roll 3d6 whenever you would roll a d20, applying bonuses and penalties normally. The possible results when rolling 3d6 (or any other multiple dice) form a bell curve- that is, a range of odds that favor average results much more than extreme results.
This system requires several changes to how rolls are made.
Automatic successes and failures: Automatic successes (for attack rolls and saves) happen on a natural 18, and automatic failures on a natural 3. Neither occurs as often as in standard d20 (less than 1/2% of the time as opposed to 5% of the time).
Taking 20 and taking 10: You can't take 20 using the bell curve variant. Instead, you have two new options: You can take 16, which makes the task take ten times as long, or you can take 18, which makes the task take one hundred times as long. As with the rules for taking 20, you can only take 16 and take 18 when you have plenty of time, when you aren't distracted, and when the task carries no consequences for failure. For a check that normally requires a standard action, taking 16 uses up 1 minute and taking 18 uses up 10 minutes.
The rules for taking ten remain unchanged.
Threat range: Because it is no longer possible to roll a natural 19 or 20, the threat ranges of weapons change in the bell curve variant. Refer to the following table.
With the bell curve variant, the narrowest threat range becomes slightly more narrow (4.6% rather than 5%) and the new 14-18 range (16%) falls between the old 18-20 and 17-20 ranges. But because the Improved Critical feat and the keen edge spell double threat ranges, characters still improve their weapons in every case, despite the flat spot on the table.
There's no table entry for a threat range of 16-20 because no combination of weapons, feats, and magic can attain it in the standard d20 rules.
Any time creatures are encountered in groups of four or more, reduce their CR by 1. For example, a single troll is CR 5, and two trolls are CR 5 each (and thus a EL 7 encounter). But four trolls are only CR 4 each (making a EL 8 encounter).
Monsters with fractional CRs move down to the next lowest fraction when encountered in groups of four or more; the goblin (ordinarily CR1/2) becomes CR 1/3, for example.
The granted power of the Luck domain changes, because simple rerolls aren't as useful in the bell curve variant as they are in the standard rules. When electing to reroll a standard result, a cleric with access to the Luck domain rolls 4d6 for the reroll (instead of 3d6), dropping the lowest die. For example, if you rolled 2, 5, 6, and 6, you would drop the 2 for a total of seventeen.
The Luck domain spells change as well, with auspicious odds replacing protection from energy at 3rd level, and mass auspicious odds replacing break enchantment at 5th level.
In general, this variant leads to a grittier d20 game, because there will be far fewer very good or very bad rolls. Not only can you no longer roll 1, 2, 19, or 20, but most rolls will be clustered around the average of 10.5. With a d20, every result is equally likely; you have a 5% chance of rolling an 18 and a 5% chance of rolling a 10. With 3d6, there is only one possible combination that results in an 18 (three sixes, obviously), but there are twenty-four combinations that result in a 10. Players used to the thrill of rolling high and the agony of the natural 1 will get that feeling less often--but it may be more meaningful when it does happen. Good die rolls are a fundamental reward of the game, and it changes the character of the game when the rewards are somewhat stronger but much less frequent.
Game balance shifts subtly when you use the bell curve variant. Rolling 3d6 gives you a lot more average rolls, which favors the stronger side in a combat. And in a d20 game, that's almost always the PCs. Many monsters--especially low-CR monsters encountered in groups-- rely heavily on a lucky shot to damage the PCs. When rolling 3d6, those lucky shots are fewer and farther between. In a fair fight in which everyone rolls a 10, the PCs should win almost every time. The bell curve variant adheres more tightly to that average (which is the reason behind the reduction in CR for monsters encountered in groups.
Another subtle change to the game is that the bell curve variant rewards bonuses relatively more and die rolls relatively less, simply because the die roll is almost always within a few points of 10. A character's skill ranks, ability scores, and gear have a much bigger impact on successes and failures than they do in the standard d20 rules.