Calculate expected averages, min/max, and standard deviation for any dice expression. Supports drop lowest, reroll, exploding dice, and modifiers.
How much damage does 2d6+4 deal on average? What's the expected ability score from 4d6-drop-lowest? The Dice Average Calculator answers these questions instantly, computing exact mathematical averages for any dice expression — no rolling required.
Enter any combination of dice count, die size, and modifier, then layer on advanced mechanics like drop-lowest, drop-highest, reroll-below-threshold, and exploding dice. The calculator provides the expected average, minimum, maximum, and standard deviation. For drop mechanics, it runs a 100,000-iteration Monte Carlo simulation for high-accuracy estimates.
This tool is indispensable for RPG character optimization, game design balance work, and probability education. Compare different weapon choices, evaluate feats, and understand exactly how much each modifier contributes to your expected outcomes. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results. Validate that outputs match your chosen standards.
Character optimization requires knowing expected values. Should you use a greatsword (2d6+4, avg 11) or a greataxe (1d12+4, avg 10.5)? Does Great Weapon Fighting favor 2d6 or 1d12 more? These questions have exact mathematical answers that our calculator provides instantly.
Game designers also need dice math for balance. Knowing that 8d6 averages 28 (with σ ≈ 4.6) tells you that ~95% of Fireballs deal 19-37 damage — critical information for encounter design.
E(NdS) = N × (1+S)/2. Var(NdS) = N × (S²−1)/12. σ = √Var. With modifier M: E = N × (1+S)/2 + M. Drop mechanics use Monte Carlo estimation.
Result: Average: 12.24, Range: 3–18, Std Dev: 2.85
The classic D&D stat generation method (4d6 drop lowest) averages 12.24 — significantly higher than 3d6's 10.5 average, making for more heroic characters.
Expected value (EV) is the average result over infinite repetitions. For a single dS: EV = (1+S)/2. For NdS: EV = N(1+S)/2. The beauty of expected value is linearity — the expected value of a sum equals the sum of expected values, regardless of whether the dice are related.
This means you can calculate E(1d8 + 2d6 + 5) = 4.5 + 7 + 5 = 16.5 by simply summing each component's expected value. No complex probability calculations needed.
While expected value tells you the center, variance and standard deviation tell you the spread. Low σ means consistent results (less swingy); high σ means wild variation. Game designers tune difficulty by choosing dice combinations that balance excitement (variance) with fairness (predictability).
The key formula: Var(NdS) = N × (S²−1)/12. This increases linearly with N (more dice) and quadratically with S (bigger dice). So 10d4 (σ ≈ 3.5) is much more consistent than 2d20 (σ ≈ 8.1) despite having a similar sum range.
"Drop lowest" mechanics shift the average upward by removing the worst result. For 4d6-drop-1, the average increases from 14 (4×3.5) to about 12.24 (from the 10.5 that 3d6 would give) — wait, it's higher than 3d6's 10.5 because you're statistically picking better dice. The exact calculation involves order statistics and is complex enough to require simulation for most practical purposes.
The average of a die is (min + max) / 2 = (1 + 6) / 2 = 3.5. While you can never roll 3.5, it's the expected value over many rolls. Over 100 rolls, your total should be close to 350.
With 100,000 iterations, the Monte Carlo estimate is typically accurate to within ±0.02 of the true average. This is more than sufficient for game balance decisions.
It measures typical spread around the average. For 2d6 (σ ≈ 2.4), about 68% of rolls fall within 7 ± 2.4 = 4.6 to 9.4. Higher σ means more swingy results.
Reroll any 1 or 2 on damage dice once. Set "Reroll if ≤" to 2. This raises 1d6 average from 3.5 to ~4.17 and 2d6 from 7 to ~8.33 — about +1.33 per d6.
2d6 averages 7 (vs 1d12's 6.5) with lower variance. You'll rarely get below 4 with 2d6, while 1d12 gives 1-3 about 25% of the time. 2d6 is more consistent and slightly higher.
Approximately 12.24. Each stat is independently rolled this way, giving an average total ability score array of about 73.5 (compared to 63 for 3d6).