Custom Dice Roller Calculator
Choose your dice type, how many to roll, and any flat modifier. The calculator instantly shows the expected value, standard deviation, minimum and maximum possible totals, and the complete probability distribution across every outcome. Perfect for tabletop RPGs, board games, statistics classes, and homebrew game design.
Formula
Worked example
Rolling 3d6+2: expected value = 3 x (6+1)/2 + 2 = 3 x 3.5 + 2 = 12.5. Standard deviation = sqrt(3 x (36-1)/12) = sqrt(8.75) = 2.96. Minimum = 3 x 1 + 2 = 5. Maximum = 3 x 6 + 2 = 20.
How the expected value and standard deviation are calculated
For a fair die with s faces, every face has equal probability 1/s. The expected value of a single roll is (s + 1) / 2, which is simply the average of 1 through s. The variance is (s^2 - 1) / 12. When you roll n identical dice, the results are independent, so their expected values add and their variances add. The expected total is n x (s + 1) / 2, and the total variance is n x (s^2 - 1) / 12. Standard deviation is the square root of variance. A flat modifier m shifts the total by m exactly, adding to the expected value without changing the spread at all.
Why the shape of the distribution matters
A single die produces a flat (uniform) distribution: every face is equally likely. When you roll two or more dice and sum them, the distribution becomes bell-shaped because there are many more ways to reach a middle total than an extreme one. With 2d6, for example, there is only 1 way to roll a 2 (both ones) but 6 ways to roll a 7. As you add more dice, the central limit theorem kicks in and the distribution becomes smoother and more tightly clustered around the mean relative to its range. This is why rolling 4d6 for character attributes in Dungeons and Dragons feels very different from rolling 1d24 even though the ranges overlap.
Modifiers, advantage, and interpreting this calculator for RPGs
A flat modifier (bonus or penalty) shifts every result by the same amount. A +3 modifier on 2d6 raises the minimum from 2 to 5 and the maximum from 12 to 15, while keeping the standard deviation identical. Many tabletop RPGs also use mechanic like advantage (roll two dice, keep the highest) or disadvantage (keep the lowest). These skewed rolls are not the same as a flat modifier and this calculator does not model them, but the expected-value and distribution outputs here give you the baseline for any dice expression written in standard notation such as 2d8+5 or 1d20-1.
Using this calculator for game design and education
Game designers use dice probability to balance difficulty: a task requiring a total of 15 on 3d6 has a very different feel from the same target on 1d20. The probability distribution chart shows you exactly how often each total appears, so you can tune difficulty curves without physical testing. Statistics students can use this tool to verify the central limit theorem in action: watch how the distribution shifts from flat to bell-shaped as you increase the number of dice from one to ten or more.
Standard polyhedral dice at a glance
| Die | Sides | Min | Max | Expected value | Std dev |
|---|---|---|---|---|---|
| D4 | 4 | 1 | 4 | 2.50 | 1.12 |
| D6 | 6 | 1 | 6 | 3.50 | 1.71 |
| D8 | 8 | 1 | 8 | 4.50 | 2.29 |
| D10 | 10 | 1 | 10 | 5.50 | 2.87 |
| D12 | 12 | 1 | 12 | 6.50 | 3.45 |
| D20 | 20 | 1 | 20 | 10.50 | 5.77 |
| D100 | 100 | 1 | 100 | 50.50 | 28.87 |
Statistics for a single die of each common type. Modifier = 0.
Frequently asked questions
What is the expected value of a dice roll?
The expected value is the long-run average of the total across many rolls. For a single N-sided die it equals (N + 1) / 2. For multiple dice it scales linearly: rolling 3d6 has an expected value of 3 x 3.5 = 10.5. A flat modifier is simply added on top. If you rolled the same dice combination thousands of times, your average total would converge to this number.
What does standard deviation tell me about a dice roll?
Standard deviation measures how spread out the results are. About 68% of rolls will land within one standard deviation of the expected value, and about 95% within two standard deviations. A high standard deviation means wild swings are common; a low one means most rolls cluster near the average. Rolling more dice shrinks the relative spread (the coefficient of variation decreases), making extreme outcomes rarer in proportion to the expected value.
How does a flat modifier affect the probability distribution?
A flat modifier shifts every possible result by the same fixed amount. Adding +3 to 2d6 raises the minimum from 2 to 5 and the maximum from 12 to 15, but the shape and spread of the distribution are unchanged. The standard deviation stays the same; only the location (mean) shifts.
What is the most likely result when rolling two D6s?
The most likely single total when rolling 2d6 is 7. There are six ways to roll a 7 out of 36 possible outcomes, giving it a 16.7% probability. Totals further from 7 become progressively less likely: 2 and 12 each have only a 1-in-36 chance (about 2.8%).
Does rolling more dice make extreme totals more or less likely?
Less likely in relative terms. Adding more dice bunches results near the center. With 1d6 every outcome has an equal 1-in-6 chance. With 10d6 the probability of rolling the minimum (10) or maximum (60) is (1/6)^10, roughly one in 60 million. Most results will land within a few points of the expected value of 35.
Can I use this for non-standard dice with unusual face counts?
Yes. Select "Custom sides" and enter any number from 2 upward. The formulas work identically for a fair die with any number of equal-probability faces. This covers unusual game dice like the D3, D5, D7, D14, and D30, as well as any custom die you create for a homebrew game.
What is RPG dice notation like "2d6+3"?
"2d6+3" means roll 2 six-sided dice and add 3 to the total. The format is NdS+M, where N is the number of dice, S is the number of sides, and M is the modifier (positive for a bonus, negative for a penalty). Omitting the modifier implies 0. This calculator generates the correct notation for you automatically based on your inputs.