Statistically, how much attack damage does the AC boost from a shield prevent on average? I've seen claims it results in a 10% reduction in damage taken, is that accurate?

# dnd-5e – Average Attack Damage Prevented by AC Boost from a Shield

armor-classdamagednd-5eshieldstatistics

#### Related Solutions

I've forgotten the formal proof for this, but hopefully this is correct:

Consider a D6 (for the sake of concrete language).

When you roll a 1, you reroll the die and keep the result. This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 2, you reroll the die and keep the result (even if it's lower). This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 3, you keep the result. This produces an average value of 3, and happens 1/6 of the time.

And so on.

This gives the following formula for the average of the D6: \$ (3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.1\bar{6}\$.

Working similar formulas for the other dice, we get this table:

\begin{array}{lccc} \hline \text{Die} & \text{(standard) Avg.} & \text{GWF Avg.} & \Delta \\ \hline \text{d4} & 2.5 & 3.00 & 0.50 \\ \text{d6} & 3.5 & 4.1\bar{6} & 0.6\bar{6} \\ \text{d8} & 4.5 & 5.25 & 0.75 \\ \text{d10} & 5.5 & 6.30 & 0.80 \\ \text{d12} & 6.5 & 7.3\bar{3} & 0.8\bar{3} \\ \hline \end{array}

Dice are independent. 2D6 will have an average value of \$2 \cdot 4.1\bar{6} = 8.3\bar{3}\$.

Common weapon average damage (Great Weapon Fighting):

\begin{array}{lcc} \hline \text{Weapon} & \text{Avg. GWF dmg} & \text{improvement w/ GWF}\\ \hline \text{Greatsword (2d6)} & 8.3\bar{3} & 1.3\bar{3} \\ \text{Greataxe (1d12)} & 7.3\bar{3} & 0.8\bar{3} \\ \text{Longsword (1d10)} & 6.30 & 0.80 \\ \text{Double-bladed Scimitar (2d4)} & 6 & 1 \\ \text{Smite (level 1, 2d8)} & 10.50 & 1.50 \\ \qquad \text{(+ weapon damage)} \\ \hline \end{array}

Observations:

The ability works out to about a +1 to damage.

It scales to almost a +3 when smiting. The more dice you add (high level smite, for example), the better the ability.

^{See errata, below}The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.

# Errata

In April of 2016, Jeremy Crawford ruled that additional dice from abilities like smite can not be re-rolled by Great Weapon Fighting.

i think that die average is just a simple divide by 2.

No, you need to add 0.5, since the damage is the average of all possible rolls. You could pair up the lowest and the highest, the second lowest and the second highest roll, ect. to get \$\frac{sides}{2}\$ pairs with sum \$sides+1\$ each. Divide the sum by the number of rolls in the pair (2) to get the average of \$(sides+1) / 2\$.

does this make a ac15 with +0 the same as an ac11 with +4?

That's the wrong way of expressing the fact you used, when calculating the number of positive outcomes. For each roll \$X\$ comparing \$X+4\$ with \$AC\$ and comparing \$X\$ with \$AC-4\$ yields the same result. Of course this means that the combinations of (AC 15, +4 attack bonus) and (AC 11, +0 attack bonus) have the same chance of hitting.

an AC 11 gives 9 out of 20 chance to hit = 9/20 = 0.45 = 45% to hit.

Not exactly. 11, 12, ..., 20 are \$10 = (20 - (AC - modifier - 1)) = 21 + modifier - AC\$ numbers.

are my calculations correct or am i missing something?

You're missing something (except for what's already mentioned): critical hits. Also you're not including your strength modifier in the damage calculation.

### Correct calculation

The expected damage without considering the increased damage of a critical hit

$$ E_{non-crit} = \frac{10}{20}\left(2\cdot\frac{1}{2}(6+1)+2\right) = \frac{90}{20} = 4.5 $$

Now we also need to double the dice, if a critical hit happens (1/20 cases); note that the base damage and the damage dice of a normal hit were already included in the previous calculation:

$$ E_{bonus\;crit} = \frac{1}{20} \cdot 2 \cdot \left(\frac{1}{2}(6+1)\right)= \frac{7}{20} = 0.35 $$

Summing both expected damage values we get \$E=4.5+0.35=4.85\$ expected damage for a single attack.

## Best Answer

## 10% is the naive answer

The +2 bonus to AC is 10% of the d20 roll, but it is more complicated than that:

^{1}.In actual games it is around 15-25%.## Calculation

Assume for simplicity that the attacker does 10 hp damage per hit, and criticals increase damage by 5 HP (50%)

^{2}.The DPR against a shieldless opponent is $$\frac{(21 - rollNeeded) * 10 + 5}{20} {}$$

The DPR against a shielded opponent is $$\frac{(21 - rollNeeded - 2) * 10 + 5}{20} {}$$ rollNeeded = (your AC) - (opponent's to hit). It is always between 2 and 20.

## Table

Substract your typical enemy's to hit from your AC, and find how much a shield would help.

For example you want to decide between dual wielding and sword-and-board for a new Fighter. As starting equipment you can get Chain mail (AC 16), and you expect many Goblins (+4 to hit). You get 12 -> 21.05% less damage received with a shield!

## Graph

1) Criticals complicate things

2) Assuming no magic items and a +5 ability, a critical increases Greatsword[GWF] by 62%, Longsword[Dueling] by 39%. Sneak Attacks gain about 88% at level 19. 50% is just an approximation, but changing it does not influence the end result significantly