Dice Average Calculator
Roll the same dice long enough and the totals settle around one number: the expected average. This calculator works that number out exactly for any roll you describe, including a flat modifier and the drop-lowest rolls tabletop games use for ability scores, and it shows the minimum, maximum, variance, and standard deviation alongside it.
Nothing here is simulated. A simulation gives you a slightly different “average” every time you run it, while the true average of a roll is a fixed mathematical fact, so this tool computes it from the formula or by counting every possible outcome.
Exact results, never simulated. Plain rolls use the closed-form formula; drop-lowest rolls are counted outcome by outcome, up to 500,000 combinations. Assumes fair dice.
Quick Answer
Key takeaways
- The average of a roll is a fixed number, not a measurement. If a tool's "average" changes between clicks, it is simulating, and you are reading sampling noise.
- Same average, different roll: 1d20 and 3d6 both average 10.500, but the d20's standard deviation is 5.766 against 2.958 for 3d6. The average tells you the center; the standard deviation tells you how wild the ride is.
- Dropping the lowest die is worth real points: it lifts 4d6 from 10.500 to 12.245 and cuts off the worst results, which is why games use it for hero characters.
- A flat modifier moves the whole distribution without stretching it. A +5 raises the average by exactly 5 and leaves variance untouched.
- Quick Answer
- Reading a roll in four numbers
- The formula behind the average
- Worked examples, from the pre-filled roll outward
- Standard dice at a glance
- Drop lowest, counted exactly
- Target numbers, and what this tool does not calculate
- What competing calculators miss
- Limits worth knowing before you rely on a number
- Frequently Asked Questions
Reading a roll in four numbers
Enter the dice, sides, drop, and modifier, or tap a preset, and the result gives you four numbers that describe the roll completely enough for most decisions.
Expected average is the long-run mean: roll the same thing hundreds of times and your running average closes in on it. It is the number to use when you compare two weapons, two spells, or two character builds.
Minimum and maximum are the hard edges of the roll. A damage roll with a minimum of 6 can finish a wounded enemy for certain; one with a minimum of 1 cannot, whatever its average says.
Variance and standard deviation measure the swing around the average, in squared units and in roll units respectively. Standard deviation is the one to read at the table: bigger means streakier.
The presets are the common cases. One tap loads 1d6, 2d6, 3d6, 4d6 drop lowest, 1d20, or 1d20 + 5 and calculates immediately, which is handy mid-game when nobody wants to watch you type.
The formula behind the average
For a fair die, every face is equally likely, so the average is the midpoint of the faces. The expected value definition in OpenStax Contemporary Mathematics, section 7.12 on expected value, weights each outcome by its probability; with equal weights that collapses to a plain average of the faces.
Run the single-die case by hand once and the rest follows: the faces of a d6 sum to 21, and 21 / 6 = 3.5. A d20 sums to 210, and 210 / 20 = 10.5. Multiple dice just add independent averages, which is why 2d6 averages 7.000 and 3d6 averages 10.500. The same linearity is what powers the next step up, a conditional expected value calculator, which averages outcomes after some of them have been ruled in or out. Drop-lowest rolls break the simple pattern, because discarding a die makes the kept dice dependent on each other, and that case is handled by counting, as the next sections show.
Worked examples, from the pre-filled roll outward
Every row below is an output of the tool above for those exact inputs, including the pre-filled 4d6 drop lowest case.
| Roll | Average | Min | Max | Variance | Std deviation |
|---|---|---|---|---|---|
| 1d6 | 3.500 | 1 | 6 | 2.917 | 1.708 |
| 2d6 | 7.000 | 2 | 12 | 5.833 | 2.415 |
| 3d6 | 10.500 | 3 | 18 | 8.750 | 2.958 |
| 1d20 | 10.500 | 1 | 20 | 33.250 | 5.766 |
| 1d20 + 5 | 15.500 | 6 | 25 | 33.250 | 5.766 |
| 4d6 drop lowest 1 | 12.245 | 3 | 18 | 8.105 | 2.847 |
The 1d20 and 3d6 rows deserve a second look, because they are the cleanest illustration of why the average alone misleads. Both center on 10.500, yet a d20 lands within a point of its average far less often than 3d6 does. The bars below plot the standard deviations the tool reports for the two rolls.
Game designers use that difference on purpose. Single-die rolls feel dramatic and stay unpredictable deep into a campaign; multi-die rolls reward the favorite and punish the long shot, because their totals bunch around the middle.
Standard dice at a glance
These are the averages the tool returns for the standard polyhedral set and the common multi-die rolls, worth bookmarking if you build characters or balance encounters often.
| Roll | Min | Max | Average |
|---|---|---|---|
| 1d4 | 1 | 4 | 2.5 |
| 1d6 | 1 | 6 | 3.5 |
| 1d8 | 1 | 8 | 4.5 |
| 1d10 | 1 | 10 | 5.5 |
| 1d12 | 1 | 12 | 6.5 |
| 1d20 | 1 | 20 | 10.5 |
| 1d100 | 1 | 100 | 50.5 |
| 2d6 | 2 | 12 | 7.0 |
| 3d6 | 3 | 18 | 10.5 |
| 4d6 drop lowest | 3 | 18 | 12.245 |
Drop lowest, counted exactly
There is no tidy closed-form formula for a drop-lowest average, so this tool does the honest thing: it lists every possible outcome, drops the lowest dice from each, and averages the kept sums.
For 4d6 that is 64 = 1,296 outcomes, a quick count, and the result is the published figure of 12.245 with variance 8.105. Notice the drop also narrows the spread compared with keeping all four dice would, and it removes the humiliating totals: a kept-three minimum of 3 requires rolling three 1s and dropping the fourth, while totals of 3 and 4 become rare instead of routine. Two honest boundaries come with the method. Counting is capped at 500,000 outcomes to keep the page fast, so a roll like 10d20 drop 1 (over ten trillion outcomes) returns a message instead of a number, rather than a silent simulation. And if you enter a drop equal to or larger than the number of dice, the tool keeps at least one die, counting the roll as drop lowest n - 1, so 2d6 with a drop of 5 is calculated as 2d6 drop lowest 1 and averages 4.472.
Target numbers, and what this tool does not calculate
The average answers "what do I roll on a typical attempt?" It does not answer "how often do I succeed?" That second question needs the count of winning outcomes over total outcomes, which is a distribution question.
The arithmetic is simple enough to do by hand for small rolls. Needing a 9 or better on 2d6: the sums 9, 10, 11, and 12 can be made in 4, 3, 2, and 1 ways, for 10 winning combinations out of 36, which is 27.78%. For the full picture of how likely every total is, a discrete probability distribution calculator lays the distribution out in full. The same counting logic sits under far simpler devices too: a coin flip probability calculator is just this tool's fair-outcome counting with two faces instead of six. On a d20 the advantage mechanic from modern tabletop rules bends the count: against a target of 11, a normal roll succeeds 50% of the time, rolling two dice and keeping the higher (advantage) succeeds 75%, and keeping the lower (disadvantage) succeeds 25%. This calculator does not model advantage or disadvantage as a mode; those figures are here so you can sanity-check a roll, not as outputs of the tool.
What competing calculators miss
The best-known rival here is good, and still stops early.
Miss 1: no drop lowest. Omni Calculator's dice average tool handles the plain formula cleanly, with the average, minimum, and maximum. It has no drop-lowest input at all, so the single most common reason players search for a dice average, the 4d6 ability-score roll, is exactly the case it cannot answer.
Miss 2: no spread. Most average-only tools, Omni included, leave out variance and standard deviation, so two rolls with the same average look identical on the page while playing completely differently at the table. This tool prints both on every result.
Miss 3: power that needs programming. AnyDice computes almost anything, drop lowest included, but you have to write function syntax to ask the question, and a syntax error is your feedback. A form with four fields and six presets answers the standard questions faster, mid-session, on a phone.
Miss 4: simulated averages. Dice rollers that average a few thousand random throws report a number that wobbles around the truth. The truth does not wobble: 3d6 is 10.500, every time, and a tool should say so.
Limits worth knowing before you rely on a number
Every formula on this page assumes fair dice with faces numbered 1 through the number of sides, which is the assumption the MathWorld entry on expected value also builds on: each outcome carries equal probability. Exploding dice, rerolls, dice pools that count successes, and advantage or disadvantage are different mechanics, and this calculator does not model them; the target-number section above shows the advantage arithmetic in prose only. Drop-lowest counting stops at 500,000 outcomes and says so rather than approximating. Loaded or badly balanced physical dice will drift from every figure here, because their faces are not equally likely, and no formula fixes that.
Frequently Asked Questions
An average is not a possible outcome; it is the center of gravity of all outcomes. Faces 1 through 6 sum to 21, and 21 / 6 = 3.5. Over many rolls your running average converges to 3.5 even though no single roll shows it.
Not always. For a single die every face is equally likely, so there is no single most common roll. For 2d6 the average is 7 and 7 is also the most common sum, but the two concepts are different: the average weights every outcome, while the most common is just the peak of the distribution.
It is the standard way to generate ability scores in Dungeons and Dragons: roll 4d6, discard the lowest die, and sum the other three. Dropping the lowest die raises the average from 10.5 to 12.245 and trims the worst outcomes.
No. A flat modifier shifts every outcome by the same amount, so the spread is unchanged. 1d20 and 1d20+5 have the same variance (33.250) and standard deviation (5.766).
How swingy the roll is. 1d20 and 3d6 both average 10.5, but the d20's standard deviation of 5.766 means wild swings, while 3d6's 2.958 clusters near the middle. If you want reliability, pick the roll with the smaller standard deviation.
Yes. Every formula here assumes each face is equally likely. Loaded or poorly balanced dice need measured probabilities, not these formulas.
Count the winning sums against all possible outcomes: for 2d6, sums of 9 through 12 have 4 + 3 + 2 + 1 = 10 winning combinations out of 36, which is 27.78%. Larger pools follow the same count-the-ways logic.
Against target 11 on a d20, a normal roll succeeds 50% of the time, advantage (roll 2d20 and keep the higher) succeeds 75%, and disadvantage (keep the lower) succeeds 25%. The math for advantage is 1 minus ((target minus 1) over 20) squared.
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References
- OpenStax, Contemporary Mathematics, "7.12 Expected Value" - the textbook definition of expected value and how to compute it from outcomes and their probabilities, the basis of every average on this page. LibreTexts: Expected Value (OpenStax Contemporary Mathematics). Accessed October 2026.
- Wolfram MathWorld, "Expected Value" - the formal definition E(x) as the probability-weighted sum of outcomes, and its properties for sums of independent variables. MathWorld: Expected Value. Accessed October 2026.
- Wolfram MathWorld, "Variance" - the definition of variance as the expected squared deviation from the mean, and the (s squared minus 1) / 12 result for a discrete uniform distribution. MathWorld: Variance. Accessed October 2026.
- National Institute of Standards and Technology, NIST/SEMATECH e-Handbook of Statistical Methods, "Measures of Scale" - how standard deviation and variance quantify the spread of a distribution around its center. NIST/SEMATECH e-Handbook: Measures of Scale. Accessed October 2026.
- Wizards of the Coast, "System Reference Document 5.1" - the official rules source describing ability score generation by rolling four six-sided dice and discarding the lowest. Wizards of the Coast: SRD 5.1. Accessed October 2026.