Learn how to calculate your D&D weapon’s average damage. Discover how to include the extra weapon damage from critical hits in the average.

## Chapter IV: The Market, Or, Sora Buys A Sword

Imagine, if you will, a bustling market-town filled with shops and stalls selling wares from every corner of the world. The acrid stink of freshly tanned hides mingles with the sickly sweet scent of apricots just past their prime. Merchant’s voices, haggling over the price of bolts of silk or kegs of mead, merge to form a fricative murmur. At the edge of town, a dragonborn — Sora Sheildbiter — stands in front of a smithy. She runs her gaze over the various weapons hung on racks along the walls, pondering her options.

The *D&D Player’s Handbook* lists 37 different weapons.^{1} Each weapon deals one of three specific damage types; bludgeoning, piercing or slashing. However, with a handful of exceptions,^{2} this damage **quality** is pure window dressing. On the other hand, damage **quantity** is a far more significant variable. With the exception of blowguns and nets, a weapon’s **base damage**^{3} is determined with dice. Smaller weapons, such as a club, deals 1D4 bludgeoning damage. And, at the other end of the scale, the aptly named greatsword deals 2D6 slashing damage. Obviously, dice have a degree of variability — for example, rolling 1D4 can yield a result of 1, 2, 3 or 4 — so it might be helpful to know the **average** amount of base damage each weapon (die) deals.

## Calculating an Average Die Roll

There are three common measures of central tendency; the **mode**, **median** and **mean**. Because the set of possible results from single die does not have a mode, we must look elsewhere for a useful measure. On the other hand, since a die roll yields **discrete data**, there is a case for using the median result. However, I happen to know a pleasant little formula for calculating the mean roll result. And who doesn’t like a pleasant little formula? So, let’s see what this *Torch of Central Tendency* illuminates.

Let be the set of possible die rolls, where is the number of dice and is the size of die, then

To see how this formula is derived, recall that for and ,

(1)

We can use this formula to find the average base damage dealt by*D&D*‘s various weapons.

And, we might be tempted to stop here, but something is missing…

## Accounting for Critical Damage

The *5th Edition* of *Dungeons & Dragons* stipulates that a weapon’s damage dice are doubled if it scores a critical hit.^{4} Which means that the greatsword **occasionally** does 4D6 slashing damage. So, how do we account for critical hits? We are going to have to drink from the *Pool of Probability* to find the answer.

There is a 1 in 20 chance^{5} of scoring a critical hit every time we roll a D20.^{6} However, we don’t just want to know the odds of scoring a crit. We want to know the odds of scoring a critical hit **assuming** we have, in fact, hit. Which means we have to take into account how likely we are to score a hit at all. In other words, We want to know the **conditional probability** of scoring a crit, given that we have hit. To put it another way, if Sora Shieldbiter hits Slobbergrot the Ogre with a greatsword, what are the chances she hits him **really** hard?

## Conditional Probability & Hitting Really Hard

Ms. Sheildbiter will only hit old Slobbergrot if her player’s attack roll (plus any applicable modifiers) is equal to, or higher than, the ogre’s armour class (AC). For simplicity’s sake, we will ignore the modifiers for the remainder of this article.

Let be the target number for a crit and be Slobbergrot’s AC. Also, let be the conditional probability of rolling a natural 20 **if** you match or beat the ogre’s AC. And, let be the probability of rolling a 20 **and** meeting or exceeding Slobbergrot’s AC. Finally, let be the probability of equaling or surpassing the ogre’s AC, then

Now, the probability of scoring a critical hit

**and**hitting Slobbergrot the Ogre is the same as the probability of scoring a critical hit.

^{7}Thus, for every AC and

For your convenience, the results of for can be found in the table below.

**Probability of a Hit**

Target Number (AC) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

1 | 0.95 | 0.9 | 0.85 | 0.8 | 0.75 | 0.7 | 0.65 | 0.6 | 0.55 | 0.5 | 0.45 | 0.4 | 0.35 | 0.3 | 0.25 | 0.2 | 0.15 | 0.1 | 0.5 |

If we plug these results into our formula for conditional probability, , we can use the results to populate the following graph.

As we can see from the graph above, the higher Slobbergrot’s AC, the smaller the chance that our hero, Sora, hits him, but the better the chance that she crits, **if she hits**.

## Weapon Damage vs. Armour Class

We would like to modify our tidy little formula for the mean to include critical damage. With this in mind, let be the probability of scoring a critical hit against a given AC, then

Now, there are twenty AC scores (1-20) and six different damage dice; (1D4, 1D6, 1D8, 1D10, 1D12 and 2D6) which yield 120 combinations. Too many to fit neatly on a single graph!

Instead, we will investigate two AC values, 6 and 14, based on *Dungeons & Dragons*‘ average armour classes. The first, is the lower end of the **effective** armour class^{8} of the average *D&D* monster.^{9} The second, is *D&D*‘s maximum effective monster AC.

With the above graph in hand, we have a good grasp of the variable average for each die type.

## Conclusions

Meanwhile, back at the smithy, Sora, continues to stare at the weapons displayed along the walls. Seeing her interest, the smith walks over and begins extolling the virtues of his weapons. He explains that the harder an opponent is to hit, the more damage his weapons do when they hit. “It really is true,” he says, “the bigger they are, the harder they fall!”

Of course, the salesman-smith hasn’t told the whole story — there is an important difference between a weapon’s **average** damage and its **expected** damage. But we will save that story for another day.

## Additional Reading

Check out these pages for more on rolling for attack damage in *Dungeons & Dragons 5E*.

- RPG Math: Calculating the Average Dice Roll — Exactly what it says on the tin, by Eldadres (a
*D&D*blogger). - DnD Weapons – A Complete Guide — A comprehensive look at
*D&D*weapons, by Jae Carbary (web developer and*D&D*aficionado).

## Notes

- The list of
*D&D*weapons are listed on page 149 of the*PHB*. - For instance, skeletons are vulnerable to bludgeoning damage.
- Sora Shieldbiter’s player may add her Strength orDexterity modifier and her proficiency bonus to an attack’s base damage.
- See page 196 of the
*PHB*for more on critical hits. - or 5%.
- Note that the Halfling’s
**Lucky**feature slightly increases the odds of rolling a critical hit. Also, certain features, such as the Champion Fighter’s**Improved Critical**extend the range for critical hits. - This is not always true for characters with features such as the Champion Fighter’s
**Improved Critical**. - A monster’s effective armour class equals its AC minus its attacker’s ability modifiers, proficiency bonus and any other applicable modifiers.
- Charmingly,
*Dungeons & Dragons*refers to potentially hostile NPCs as**monsters**.