If you can calculate how a feat, a weapon, a spell, a tactic, or a magic feature affects your performance in combat, you can better select those options that help you most.

Average Damage per Round
A key statistic to know is your average (mean) damage per round. Basically, you multiply the number of successful hits you get in the average round by the amount of damage you deal per hit. (Since the number of hits you can expect per round varies by the opponent’s AC, you need to look at different ACs over a range of typical ACs.)

Example: At 1st level, Tordek the dwarven fighter attacks at +4 for 1d10+2 damage. His average damage vs. AC 16 (an ogre) is 45% (chance to roll 12+ on 1d20) times 7.5 (average damage) for a total of 3.375. His average damage vs. AC 11 (a zombie) is 70% ´ 7.5 = 5.25.

Warning: When you calculate average damage this way, you assume that every point of damage you deal counts. Against a kobold with 2 hit points, Tordek’s average damage with a successful hit is 3 points. Any more damage than that is wasted. Thus, he deals the same average damage per hit with his fist as with a dwarven waraxe. When you’re fighting creatures with fewer hit points than you can deal damage, not all of your damage capacity "counts."

Criticals and Average Damage
Think of a standard critical as "a 5% chance to deal +100% damage," which averages out to +5% damage.

A swordlike critical (threat on 19 and 20) is "a 10% chance to deal +100% damage," which averages out to +10% damage.

An axelike critical (triple damage) is "a 5% chance to deal +200% damage," which also averages out to +10% damage.

Thus, Tordek’s average damage is actually 3.7125 per round vs. ogres. His triple criticals don’t improve his damage vs. zombies because undead are not subject to critical hits.

Attack Stats
Sometimes you need to choose between more attacks, higher attack bonuses, and more damage. For instance, a monk can use a "flurry of blows" to get one extra attack at the cost of suffering –2 on all attack rolls. The Power Attack feat also lets you decrease your attack bonus to increase your damage. We can compare how different benefits affect average damage.

Example: Ember, a 1st-level monk, is fighting a monster with AC 14 (such as an orc). She can strike with one unarmed attack (attack +2) or with two, as a flurry (attack +0). Her number of expected hits per round is .45 with one attack (her chance to roll 12+ on 1d20), and her number of expected hits per round is .7 with two attacks (35% chance to roll 14+, twice). Wrinkle: If Ember is fighting an orc, there’s a 23% chance that she won’t have an opportunity for a second attack (because the first attack does the orc in). Thus, her chance to hit with the second attack is really only 77% ´ 35%, or 27%. Thus her expected number of hits against the orc is .35 + .27 or .62 (still better than .45 with one attack.)

Example: At 1st level, the human fighter Regdar (attack +4, damage 11 with a two-handed sword [damage 10 vs. creatures not subject to criticals]) has Power Attack. Against low AC creatures, it’s a good idea to use it. Against high AC creatures, it’s a bad idea. This table shows Regdar’s average expected damage against these creatures.

	Gelatinous Cube (AC 3)	Zombie (AC 11)	Ogre (AC 16)	Ogre in Half Plate (AC 20)
Normal Attack	9.5	7.0	5.0	2.8
Power Attack (–1/+1)	10.5	7.2	4.8	2.4

Randomness vs. Predictability.
Raw calculations of averages doesn’t take into account randomness. Player characters want to reduce randomness. Randomness favors the underdog, and PCs are usually the favored side in a fight. Thus, the more randomness there is in a fight, the worse it is for the PCs.

Hypothetical: Imagine a monster that dealt 1 damage to one character every round and another that had a 1 in 20 chance to deal 20 damage to one character every round. Their average damage would be the same, but the one capable of getting lucky and dealing 20 damage all at once is much more likely to kill a 1st-level PC.

Orcs: Orcs have only about 4 hit points, and their AC is only 14, but they wield greataxes that deal 1d12+3 damage with triple damage criticals. They don’t last long, and their average damage isn’t that bad, but if you fight orcs several times, one of them is bound to get lucky eventually and hit a character with a critical (average damage 28.5).

Offense vs. Defense

Sometimes it pays to improve your attack at the expense of your defense (such as by charging). Sometimes it pays to improve your defense at the expense of your attack (such as with Expertise).

Example: Tordek’s facing a goblin. Should Tordek charge (+2 attack, –2 AC) or attack normally?

Stats
Tordek: +4 attack, AC 17, d10+2 damage
Goblin: +0 attack, AC 15, hp 5
Combat
Tordek chance to drop, normal: chance to hit ´ chance to deal 5 damage.
	= 50% x 80% = 40%
Tordek chance to drop, charge:
	= 60% x 80% = 48%
Goblin chance to hit Tordek, standard:
	= 20%
Goblin chance to hit Tordek, charge:
	= 30%
Chance for goblin to survive Tordek’s attack and hit him.
	If Tordek attacks normally: 60% x 20% = 12%
	If Tordek charges: 52% x 30% = 15.6%
Tordek is fighting the goblin. Should he fight normally or defensively (–4 attack, +2 AC)?
	Tordek’s chance to hit the goblin, normal/defensive: 50%/30%
	Goblin’s chance to hit Tordek, normal/defensive: 20%/10%
	Fighting normally, Tordek hits the goblin two and one-half times as often as the goblin hits him. Fighting defensively, Tordek hits the goblin three times as often as the goblin hits him.
	Fighting defensively has a hidden benefit. The longer a fighter can drag out a combat, the more spells the spellcasters will have time to cast.

About the Author

Jonathan Tweet, a senior designer at Wizards of the Coast, led the 3rd Edition D&D design team and authored the new Player’s Handbook. Since 1986, he has freelanced, self-published, and worked full time in the adventure game industry. His other design credits include Ars Magica, Over the Edge, Everway, and support material for AD&D, Magic: The Gathering, Netrunner, RuneQuest, and Talislanta.