Dice Average Calculator

Dice average formula

Every face of a fair die is equally likely, so its average is the midpoint of its lowest and highest values: (1 + Y) ÷ 2. For several dice, multiply that by the number of dice; a flat modifier is simply added on. For 3d6 + 2: 3 × 3.5 + 2 = 12.5.

This works because the expected value of a sum equals the sum of the expected values, even when the dice are different. 2d6 + 1d8 + 3 averages 7 + 4.5 + 3 = 14.5.

Average of every standard die

Standard deviation: how far rolls stray from the average

The average tells you where results centre; the standard deviation (SD) tells you how widely they scatter. One die has a variance of (Y² − 1) ÷ 12, and variances add for independent dice, so the SD of XdY is √(X × (Y² − 1) ÷ 12).

Compare 2d6 with 1d12. They have almost the same average (7 versus 6.5), but 2d6 has an SD of 2.42 and 1d12 an SD of 3.45. 2d6 clusters around 7, while 1d12 is equally likely to roll a 1 or a 12. More dice means a tighter bell-shaped curve: with 3d6, about 68% of rolls land between 8 and 13, and a 3 or an 18 each comes up only once in 216 rolls.

D&D 5e examples

Average damage

On a critical hit in 5e you roll the damage dice twice but add the modifier once, so the average rises by the dice part only: a 2d6+4 greatsword goes from 11 to 18.

Expected damage per attack

To compare weapons fairly, multiply average damage by your chance to hit. With +5 to hit against AC 15 you need a 10 or more on the d20 – 11 faces out of 20, or 55%. A 2d6+4 attack therefore averages about 0.55 × 11 = 6.05 damage per swing, before adding the small extra from critical hits (5% chance of +7).

Advantage and disadvantage

Rolling two d20s and taking the higher (advantage) raises the average from 10.5 to 13.825; taking the lower (disadvantage) drops it to 7.175. That is a shift of about 3.3 points, which is why advantage is often described as worth roughly +3 to +5 depending on the target number.

Ability scores and hit points

The common 4d6-drop-lowest method for ability scores averages about 12.24, compared with 10.5 for a straight 3d6. For hit points after first level, 5e lets you take a fixed value instead of rolling: 4 for a d6, 5 for a d8, 6 for a d10 and 7 for a d12 – each half a point above the true average, rounded up.

Average versus most likely result

With one die every result is equally likely, so there is no most likely value. With two or more dice, totals near the average are the most common – 7 is the most likely total on 2d6, with a 1-in-6 chance. With an odd number of dice the average can fall between two whole numbers; on 3d6 both 10 and 11 share the top spot at 12.5% each.

Dice notation explained

Tabletop games write rolls as XdY + Z. X is how many dice you roll, Y is how many sides each die has and Z is a flat number added or subtracted afterwards. So 3d8 − 1 means roll three eight-sided dice, add them together and subtract one. A leading number can be left out when it is one: d20 means 1d20. Percentile dice, written d100 or d%, are usually rolled as two ten-sided dice – one for the tens and one for the units – and average 50.5.

Some systems add extra notation: 4d6kh3 or 4d6dl1 means roll four d6 and keep the highest three; 2d20kh1 is advantage and 2d20kl1 is disadvantage. The calculator accepts these short forms too.

How to calculate the average dice roll by hand

1. Split the expression into dice groups and modifiers: 2d6 + 1d8 + 3.
2. Find each die's average with (sides + 1) ÷ 2: d6 = 3.5, d8 = 4.5.
3. Multiply by the number of dice in each group: 2 × 3.5 = 7 and 1 × 4.5 = 4.5.
4. Add the groups and the modifier: 7 + 4.5 + 3 = 14.5.
5. For the range, the minimum is one per die plus the modifier (3 + 3 = 6) and the maximum is every die on its top face plus the modifier (12 + 8 + 3 = 23).

Probability of each total on 2d6

The triangle shape is why 6, 7 and 8 come up so often in board games that use two dice, such as Catan, and why 2 and 12 are rare. Add a third die and the triangle smooths into a bell curve.

Rerolls and exploding dice

Some rules change the average by letting you reroll. With the 5e Great Weapon Fighting style, you reroll a 1 or 2 on a damage die once and keep the new roll. That lifts a d6 from 3.5 to about 4.17 and a d12 from 6.5 to about 7.33, so a 2d6+4 greatsword averages about 12.3 instead of 11.

Exploding dice – roll again and add whenever a die shows its top face, as in Savage Worlds – raise the average further. An exploding d6 averages 3.5 × 6 ÷ 5 = 4.2, and an exploding dY in general averages (Y + 1) ÷ 2 × Y ÷ (Y − 1).

Using averages to balance a game

Game designers often compare a monster's hit points with the average damage a party can deal per round to estimate how many rounds a fight will last. A creature with 45 hit points facing four characters who each average 9 damage per round lasts about 45 ÷ 36 = 1.25 rounds if every attack hits – and closer to two or three rounds once misses are counted. Averages give a starting point; the standard deviation tells you how often a fight will end much sooner or later than expected.

Averages are also handy at the table. A game master who uses the average damage printed in a monster's stat block (shown in brackets before the dice, for example 13 (2d8 + 4)) keeps the game moving without slowing down to roll every hit.

Sources

1. Wizards of the Coast – D&D Basic Rules / System Reference Document 5.1 (critical hits, hit points, advantage) — dndbeyond.com/srd