Last Updated: September 26, 2021
Everyone wants to do more damage. In a game where one point of damage is the difference between getting a turn and lying on the ground dying, dealing a few more points of damage is important. But simply adding a damage bonus to your attacks doesn’t give us a complete picture, and choosing between a bonus to attacks or a bonus to damage is mostly guess work.
The purpose of this article is to establish a mathematically robust way to calculate “damage per round” (DPR) that takes into account the games various mechanics so that we can make objective, factual comparisons when weighing decisions with unclear results.
This article establishes a “DPR Score”. DPR indicates the average amount of damage a single character can deal over the course of a single round assuming no action or intervention from other creatures or from other outside forces.
If you’re not interested in the explanation of the logic behind the DPR calculation, or if you don’t enjoy algebra and statistics, I won’t blame you for backing out of this article. This stretches the limits of my understanding of statistics, and
The simplest part of DPR is the character’s attack and damage rolls, and they’re the most important part of the equation. If a character’s attack bonus is terrible, their DPR will suffer greatly because they can’t hit anything. If the character’s attack bonus is great but their damage is poor, their DPR will suffer. This establishes a crucial balance between attack bonus and damage which is central to the DPR score.
We will define our attack bonus as “A”, for “Attack”. Nice and simple.
We will define the AC of a hypothetical enemy as “M”, for “Monster”. Determine this value by check the AC of a creature of the character’s level according to the table “Monster Statistics by Challenge Rating” found on page 274 of the Dungeon Master’s Guide.
We will define the value of our damage as “D”, for “Damage”, where D is the total of the value of the average roll of each damage die dealt by the character’s attacks. Static damage bonuses like that from Strength added to a weapon attack are omitted because those bonuses are not multiplied on a critical hit. This can include many sources of bonus damage: Hunter’s Mark, Hex, Sneak Attack, Divine Smite, etc. are all multiplied. For clarification, see this Sage Advice twitter thread from Jeremy Crawford.
For convenience, I’ve included a table of the average values of each die.
For Example: Craig the 1st-level Warlock casts Hex on a creature, then attacks it with a Glaive. Craig normally deals 1d10+3 damage with his Glaive, and adds +1d6 for the bonus damage from Hex. 1d10 averages to 5.5 damage, and 1d6 averages to 3.5 damage. Adding both dice (5.5 + 3.5) gives us a total of 9, which is our value for D. Remember that static bonus (Craig’s +3 bonus) are not added to the value of D because they are not multiplied on a critical hit.
Any damage not multiplied on a critical hit becomes “B”, for “Bonus”. This includes things like Strength damage to weapon attacks, bonus damage from the Fighting Style (Dueling), etc.
Critical hits complicate our damage calculations, and this is further complicated by the availability of items and class features which allow characters to score critical hits on rolls other than a natural 20. However, since we separate our damage dice and our other damage bonuses, we can easily include critical hits in our equation.
We will define a variable for our critical hits, C, for “Critical Hit”, which is taken from the table below:
We will define our hit chance as “H”, for “Hit”. Hit will be a value ranging from 0.5 to 0.95, which is the probability of a hit represented as a decimal. Probabilities typically range from 0 (impossible) to 1 (guaranteed), but natural 1’s always miss and natural 20’s always hit.
H = 1 – ((M – A)/20)
Advantage / Disadvantage
Advantage and Disadvantage are a crucially important mechanic in 5e. Some characters can easily gain Advantage (Barbarians, Kobolds with Pack Tactics, anyone with a familiar, etc.) on one or more attacks, potentially leading to a significant increase in DPR.
We’ll define our hit change with Advantage as HA and our hit chance with Disadvantage as HD. Like our regular Hit Chance, natural 1’s and 20’s factor into the equation, giving both HA and HD a minimum of 0.0025 and a maximum of 0.9975.
H = 1 – ((M – A – 1)/20) HA = 1 – ((M – A)/20)^2 HD = ((20 – M + A)/20)^2
We also need to consider how Advantage and Disadvantage affect critical hits and their effect on our DPR. We’ll define our critical change with Advantage as CA and our critical chance with Disadvantage as CD.
CA = 1 – (1-C)^2 CD = C^2
DPR = C * D + H * (D + B) DPRA = CA * D + HA * (D + B) DPRD = CD * D + HD * (D + B)