Understanding the DnD Dice Roller
Tabletop games like Dungeons & Dragons describe every roll in "dice notation": NdX+M, where N is the number of dice, X is the number of sides on each die, and M is a flat modifier added to (or subtracted from) the total. So 3d6+2 means "roll three six-sided dice, add them together, then add 2." This calculator does not simulate a single random roll — it reports the exact statistics of the whole roll: the smallest and largest totals possible, the long-run average, and how much the total typically bounces around.
The formulas
A fair X-sided die is uniform over the values 1 through X. From that, every result on this page follows directly:
- Minimum total: N + M — every die shows its lowest face (a 1), so the smallest sum is N, plus the modifier.
- Maximum total: N·X + M — every die shows its highest face (X).
- Average (expected value): N·(X+1)/2 + M. A single die averages (X+1)/2 — for a d6 that is 3.5 — and averages add across dice, then the modifier is added once.
- Standard deviation: √( N·(X²−1)/12 ). Each fair die has variance (X²−1)/12; because the dice are independent, variances add, and the standard deviation is the square root of the total. The flat modifier shifts the average but never changes the spread.
Why this is useful
Knowing the average tells a designer or player whether a d10 (avg 5.5) versus 2d6 (avg 7) is the stronger damage die, and the standard deviation tells you how swingy it is. A single d12 (avg 6.5, SD ≈ 3.45) and 2d6 (avg 7, SD ≈ 2.42) are close in average but the 2d6 result is far more consistent — the more dice you roll, the tighter the total clusters around the mean. That trade-off between average and reliability is the core reason to compute these numbers instead of guessing.
Common reference points
- d20: average 10.5, min 1, max 20, SD ≈ 5.77 — the classic attack/check die.
- d6: average 3.5, SD ≈ 1.71.
- 3d6: average 10.5, min 3, max 18, SD ≈ 2.96 — the traditional ability-score roll, tightly centered on 10–11.
- 4d6 drop lowest: not computed by the simple formula above (it is not a plain sum), but averages about 12.24 — worth knowing because this page assumes you keep every die.
Frequently Asked Questions
What is the average of 3d6?
Does adding a modifier change how swingy the roll is?
Why does the calculator not handle "4d6 drop lowest"?
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|---|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|---|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|---|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |