Dice Average Calculator
Free dice average calculator. Expected value, variance, SD for 2d6, 8d6, mixed dice. D&D damage, Fir
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Expected Value, Variance, Distribution for Dice Rolls
E(X), Var(X), SD for any dice combination. D&D, Pathfinder, tabletop RPG. PMF chart, per-die breakdown, notation parser.
Quick notation (e.g., 2d6+1d8+3)
Real-World Scenarios โ Click to Load
Dice Groups (1โ5)
Per-Die Breakdown
| Group | Expected | Variance |
|---|---|---|
| 2d6 | 7.0000 | 5.8333 |
Probability Distribution
Per-Die Expected Value
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- โข Expected value of a single d6 = 3.5 (not 3 or 4!)
- โข 2d6 averages 7 but has a bell curve; 1d12 averages 6.5 but is flat โ same range, different shapes
- โข Adding more dice makes the distribution more "normal" (Central Limit Theorem)
- โข Modifiers shift the average by exactly the modifier amount without changing the spread
- โข Variance of n identical dice = n ร (fยฒ-1)/12
- โข E(total) = nยท(f+1)/2 + modifier for ndf + modifier
Did You Know?
Expert Tips
2d6 vs 1d12
Same range (2-12), but 2d6 favors the middle โ better for consistent damage
Modifiers
+5 to hit shifts average by 5 but doesn't change variance โ spread stays the same
Fireball 8d6
Average 28, SD โ4.8 โ most rolls fall between 23 and 33
Notation
2d6+1d8+3 means two d6, one d8, plus 3 modifier
How It Works
1. Single Die Expected Value
E(X) = (f+1)/2. For a d6: (6+1)/2 = 3.5. The average of 1,2,3,4,5,6.
2. Variance and Standard Deviation
Var(X) = (fยฒ-1)/12. For n independent dice, variances add. SD = โVar.
3. Convolution for PMF
For small combinations, we convolve die distributions to get the exact probability mass function.
4. Normal Approximation
For large dice pools (>10000 outcomes), the Central Limit Theorem gives a normal approximation.
Why Use This Calculator vs. Other Tools?
| Feature | This Calculator | AnyDice | Manual |
|---|---|---|---|
| Mixed dice + modifier | โ | โ | โ ๏ธ |
| PMF bar chart | โ | โ | โ |
| Per-die breakdown | โ | โ ๏ธ | โ |
| Notation parser | โ | โ | โ |
| Copy & share results | โ | โ | โ |
| AI-powered interpretation | โ | โ | โ |
| Educational content | โ | โ | โ |
Frequently Asked Questions
What is the expected value of 1d6?
3.5 โ the average of 1, 2, 3, 4, 5, 6. Not 3 or 4!
Why does 2d6 average 7?
Each d6 averages 3.5. 3.5 + 3.5 = 7. The distribution is bell-shaped, favoring 7.
Do modifiers change variance?
No. Adding a constant shifts the mean but variance stays the same.
What notation does the calculator accept?
Standard RPG notation: 2d6+1d8+3 means two d6, one d8, plus 3.
When is the normal approximation used?
When total outcomes exceed 10,000 (e.g., 9+ d6).
Why 2d6 vs 1d12 for damage?
2d6 has the same range (2-12) but clusters around 7 โ more consistent.
What is the difference from Dice Probability Calculator?
Dice Average gives E(X), variance, and full distribution. Dice Probability gives exact P(sum=k) for specific targets.
How do I interpret the PMF chart?
Each bar shows P(total = that value). The distribution shows how likely each outcome is.
Dice by the Numbers
Official Data Sources
Disclaimer: This calculator provides dice probability for educational and tabletop RPG reference. For large dice pools, the normal approximation is used. Exact PMF is computed via convolution when total outcomes โค 10,000.
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