Dice Average Calculator
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Enter the probability of landing on each face. The probabilities must add up to 1.
Run a calculation in the Dice Average, Simulate Rolls, or Biased Die tab first, then return here to see the full step-by-step solution.
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What Is the Average of a Die?
For a fair die numbered 1 through s, the average of every possible outcome is (1 + s) / 2. For a standard six-sided die, that works out to (1 + 6) / 2 = 3.5. The average is not a value the die can actually land on. It is the long-run mean you would expect if you rolled the die many times and averaged the results.
This shortcut works because the outcomes on a fair die are evenly spaced integers. When numbers are evenly spaced, the midpoint of the lowest and highest values equals the arithmetic mean of the whole set. That is exactly what (1 + s) / 2 calculates: the midpoint between 1 and s. According to Wolfram MathWorld's definition of a fair die, this property depends on every face having an equal chance of appearing.
Dice Average Formula
For a fair die with outcomes 1 through s, the expected value is E(X) = (1 + s) / 2. For n identical dice added together, the expected total is E(total) = n × (1 + s) / 2.
Average of One Die
Average = (1 + s) / 2
Where:
s = number of sides
Outcomes run from 1 to s
Average Total of n Dice
Average total = n × (1 + s) / 2
Where:
n = number of dice
s = sides per die
Both formulas assume the dice are fair (every face equally likely) and rolled independently, meaning the result of one die has no effect on any other die. Those two assumptions hold for ordinary dice used in board games, tabletop role-playing games, and craps.
Why Is the Average of a d6 3.5?
Add every possible result on a six-sided die (1+2+3+4+5+6 = 21) and divide by the number of faces (6). 21 ÷ 6 = 3.5.
A single roll of a d6 can only land on 1, 2, 3, 4, 5, or 6. It can never show 3.5. That number describes what happens over many rolls, not what happens on any one roll. Beginners often expect the average to be a real outcome, but the average of a discrete set of numbers does not have to be one of the numbers in the set.
Average of Common Dice
The average result scales directly with the number of sides on the die. The table below lists the minimum, maximum, and average for every standard die, assuming each die is fair and numbered from 1 to its highest face.
Table: Average of a Single Standard Die
| Die | Minimum | Maximum | Average |
|---|---|---|---|
| d2 | 1 | 2 | 1.5 |
| d4 | 1 | 4 | 2.5 |
| d6 | 1 | 6 | 3.5 |
| d8 | 1 | 8 | 4.5 |
| d10 | 1 | 10 | 5.5 |
| d12 | 1 | 12 | 6.5 |
| d20 | 1 | 20 | 10.5 |
| d100 | 1 | 100 | 50.5 |
Average of Multiple Dice
Averages add across independent dice. The average total of n identical dice equals n times the average of a single die.
2d6
2 × 3.5 = 73d6
3 × 3.5 = 10.54d6
4 × 3.5 = 145d6
5 × 3.5 = 17.5This holds no matter how many dice are involved, and it works even when the dice have different numbers of sides. Add the average of each die individually to get the average of the total.
Common Dice Combinations and Their Averages
Table: Dice Notation, Range, and Average
| Notation | Dice | Sides | Minimum | Maximum | Average |
|---|---|---|---|---|---|
| 1d4 | 1 | 4 | 1 | 4 | 2.5 |
| 2d4 | 2 | 4 | 2 | 8 | 5 |
| 3d4 | 3 | 4 | 3 | 12 | 7.5 |
| 1d6 | 1 | 6 | 1 | 6 | 3.5 |
| 2d6 | 2 | 6 | 2 | 12 | 7 |
| 3d6 | 3 | 6 | 3 | 18 | 10.5 |
| 4d6 | 4 | 6 | 4 | 24 | 14 |
| 1d8 | 1 | 8 | 1 | 8 | 4.5 |
| 2d8 | 2 | 8 | 2 | 16 | 9 |
| 3d8 | 3 | 8 | 3 | 24 | 13.5 |
| 1d10 | 1 | 10 | 1 | 10 | 5.5 |
| 2d10 | 2 | 10 | 2 | 20 | 11 |
| 1d12 | 1 | 12 | 1 | 12 | 6.5 |
| 2d12 | 2 | 12 | 2 | 24 | 13 |
| 1d20 | 1 | 20 | 1 | 20 | 10.5 |
| 2d20 | 2 | 20 | 2 | 40 | 21 |
| 3d20 | 3 | 20 | 3 | 60 | 31.5 |
| 1d100 | 1 | 100 | 1 | 100 | 50.5 |
| 2d100 | 2 | 100 | 2 | 200 | 101 |
What Does 2d6 Mean?
Dice notation follows the pattern NdS, where N is the number of dice and S is the number of sides on each one. 2d6 means roll two six-sided dice and add the results. 1d20 means roll a single 20-sided die.
1d20
One 20-sided die2d6
Two six-sided dice, added together4d8
Four eight-sided dice, added together2d6 + 3
Two six-sided dice, plus a fixed +3 modifierHow to Calculate the Average of a Die
Add the lowest and highest possible results, then divide by 2. This shortcut works for any fair die numbered from 1 to s, because the outcomes are evenly spaced.
For a standard die, this is 1.
This equals the number of sides, s.
1 + s
Average = (1 + s) / 2. For n dice, multiply this result by n.
Average per Die vs. Average Total
The average of a single die and the average of a total made from several dice are two different numbers, and mixing them up is a common source of confusion.
Table: 2d6 Average per Die vs. Average Total
| Measure | Value | Meaning |
|---|---|---|
| Average per die | 3.5 | The long-run average of one d6 on its own |
| Average total (2d6) | 7 | The long-run average of the sum of both dice |
Worked Dice Average Examples
Example 1: One d6
(1 + 6) / 2
3.5
Example 2: Two d6
3.5
2 × 3.5 = 7
7 is also the most common single total when rolling 2d6, since six of the 36 equally likely dice combinations add up to 7, more than any other sum.
Example 3: Three d6
3 × 3.5 = 10.5
Minimum 3, maximum 18
Example 4: One d20
(1 + 20) / 2
10.5
Example 5: Two d20
2 × 10.5 = 21
Minimum 2, maximum 40
Example 6: One d100
(1 + 100) / 2
50.5
Dice Average and Expected Value
For a fair die, the average and the expected value are the same number. Expected value is a probability-weighted average: multiply each possible outcome by its probability and add the results. On a fair die, every face has the same probability, 1/s, so the weighted average simplifies to the plain arithmetic mean, (1 + s) / 2. The two ideas only diverge once the probabilities stop being equal, which is the case covered next.
Fair vs. Biased Dice
The (1 + s) / 2 shortcut assumes a fair die, meaning every face has an equal chance of landing face up. A biased or loaded die does not follow this rule. Its expected value is calculated with the general formula E(X) = Σ x × P(X = x), the sum of each outcome multiplied by its own probability.
1(0.10) + 2(0.10) + 3(0.15) + 4(0.15) + 5(0.20) + 6(0.30)
0.10 + 0.20 + 0.45 + 0.60 + 1.00 + 1.80 = 4.15
The expected value of this biased die is 4.15, noticeably higher than the fair-die average of 3.5. Use the Biased Die tab above to try your own probabilities.
Theoretical Average vs. Observed Average
The theoretical average is the long-run expected result calculated from the probability distribution. The observed average is whatever a specific set of real or simulated rolls actually produced.
Roll a d6 ten times and the observed average might land at 4.1, or 2.8, or almost anywhere else in a reasonable range. The theoretical average stays 3.5 regardless. Small samples fluctuate; that is expected, not a sign that something is wrong with the die.
Why Does the Observed Average Approach the Expected Value?
As the number of independent rolls grows, the observed average tends to move closer to the theoretical expected value. This pattern is known as the law of large numbers. It describes a tendency, not a guarantee: there is no fixed number of rolls after which the observed average is required to match the theoretical one exactly, and earlier unusual results are never "corrected" by later rolls. Each roll remains completely independent of the ones before it. You can see this pattern directly using the Simulate Rolls tab above. Try running 50 rolls, then 5,000, and compare how far the observed average drifts from the theoretical value each time.
Is the Average the Most Likely Dice Result?
Sometimes, but not always. For a single fair die, every face is equally likely, so there is no single most likely result at all, even though the average is a specific number like 3.5. Once you add multiple dice together, some totals become more common than others, because there are more ways to reach a middle value than an extreme one.
Table: 2d6 Totals, Combinations, and Probability
| Total | Combinations | Probability |
|---|---|---|
| 2 | 1 | 2.78% |
| 3 | 2 | 5.56% |
| 4 | 3 | 8.33% |
| 5 | 4 | 11.11% |
| 6 | 5 | 13.89% |
| 7 | 6 | 16.67% |
| 8 | 5 | 13.89% |
| 9 | 4 | 11.11% |
| 10 | 3 | 8.33% |
| 11 | 2 | 5.56% |
| 12 | 1 | 2.78% |
There are 36 equally likely combinations for 2d6, and 7 has the most ways to occur (six of them: 1-6, 2-5, 3-4, 4-3, 5-2, 6-1). That is why 7 is both the average and the most common total for 2d6. For a single die, or for dice with very different numbers of sides, the average and the most common outcome frequently do not line up at all.
Mean, Median, and Mode of Dice Rolls
Table: Mean, Median, and Mode for One d6 and 2d6
| Concept | Meaning | One d6 | 2d6 |
|---|---|---|---|
| Mean | Probability-weighted average | 3.5 | 7 |
| Median | Middle value of the distribution | 3.5 | 7 |
| Mode | Most frequently occurring result | None (all equally likely) | 7 |
Dice Average With Modifiers
A constant modifier shifts the expected value by exactly that amount. It does not change how the dice themselves behave, only the total.
2d6 + 3
7 + 3 = 101d20 + 5
10.5 + 5 = 15.5Dice Average With Multiplication
Multiplying a single die's result is different from rolling several dice and adding them, even when the average comes out the same.
2d6: roll two separate d6 and add them. Average = 7 as well, but every integer from 2 to 12 is possible, and the totals cluster around 7 far more than the doubled single-die version does.
Both approaches share the same average, but they produce very different spreads of outcomes. This is why variance and standard deviation, not just the average, matter when comparing dice mechanics.
Dice Average Formula for Different Dice
When dice have different numbers of sides, add each die's individual average separately.
General Formula
E(total) = Σ (s₀ + 1) / 2
For each fair die numbered
1 through its own s value1d6 + 1d8
3.5 + 4.5 = 8Advanced: Keeping the Highest or Lowest of Multiple Dice
The ordinary dice-average formula does not apply when you roll extra dice and keep only the best (or worst) result. That mechanic changes the underlying distribution, so the expected value has to be calculated from the distribution of the maximum or minimum, not from a simple average.
For two independent dice numbered 1 through s, the expected value of the higher result is (s + 1)(4s − 1) / (6s), and the expected value of the lower result is (s + 1)(2s + 1) / (6s). The two always add up to s + 1, which matches the fact that a die's minimum and maximum results are symmetric around its average.
Table: Rolling Two d20 and Keeping One Result
| Mechanic | Average | Compared to a Single d20 (10.5) |
|---|---|---|
| Single d20 | 10.5 | Baseline |
| 2d20, keep the higher result | 13.825 | +3.325 |
| 2d20, keep the lower result | 7.175 | −3.325 |
Other mechanics such as rerolling low results, dropping the lowest die from a larger pool, or exploding dice (rolling again and adding whenever the maximum face appears) shift the average as well, but the exact amount depends entirely on the specific rule being used. Each one requires its own probability-weighted calculation rather than a single fixed formula.
Related Topics and Calculators on Statistics Fundamentals
Dice averages connect directly to expected value, probability distributions, and simulation. These resources build out the full picture.
Sources and Further Reading
References used in this guide:
- Weisstein, Eric W. Dice. From MathWorld, a Wolfram Resource. mathworld.wolfram.com
- Weisstein, Eric W. Fair Dice. From MathWorld, a Wolfram Resource. mathworld.wolfram.com
- Khan Academy. Random Variables and Expected Value. khanacademy.org
- Law of Large Numbers. Wikipedia. en.wikipedia.org
- Expected Value. Wikipedia. en.wikipedia.org
Frequently Asked Questions
The average roll of a standard six-sided die is 3.5. Add the six possible outcomes (1+2+3+4+5+6 = 21) and divide by 6. No single roll produces 3.5; it is the long-run average across many rolls.
The average roll of a 20-sided die is 10.5, using (1 + 20) / 2. The minimum result is 1 and the maximum is 20.
The average roll of a 100-sided die, or a percentile roll built from two ten-sided dice, is 50.5, using (1 + 100) / 2.
The average total of two six-sided dice is 7. Each die averages 3.5, and averages add across independent dice: 3.5 + 3.5 = 7. Seven is also the most common total for 2d6, since six of the 36 equally likely combinations add up to 7.
The average total of three six-sided dice is 10.5, calculated as 3 × 3.5. The minimum possible total is 3 and the maximum is 18.
For a fair die numbered 1 through s, add the lowest and highest possible results and divide by 2: average = (1 + s) / 2. This works because the outcomes are evenly spaced, so the midpoint of the range equals the arithmetic mean of all outcomes.
For a fair die, yes. Expected value is a probability-weighted average, and every face on a fair die has the same probability, so the expected value equals the simple arithmetic mean, (1 + s) / 2. On a biased die the two can differ, since expected value then depends on the actual probability of each face.
No. A single roll of a six-sided die can only land on 1, 2, 3, 4, 5, or 6. The value 3.5 describes the theoretical long-run average of many rolls, not an outcome of any individual roll.
The average of several independent dice added together equals the sum of each die's individual average. For n identical dice with s sides, the average total is n × (1 + s) / 2. For dice with different sides, add each average separately, for example 1d6 + 1d8 averages 3.5 + 4.5 = 8.
No. For a single fair die, every face is equally likely, so there is no single most likely result even though the average is 3.5. For multiple dice added together, the average and the most common total often do match, for example both equal 7 for 2d6, because middle totals have more combinations than extreme ones.
The simple formula (1 + s) / 2 only holds when every face is equally likely. For a biased die, the expected value is the probability-weighted sum of the outcomes: multiply each face by its own probability and add the results. A die weighted toward higher numbers will have an expected value above (1 + s) / 2.
Individual rolls are random, so a small number of rolls can land noticeably above or below the theoretical average just by chance. As the number of independent rolls grows, the observed average tends to move closer to the theoretical value, a pattern described by the law of large numbers. It does not guarantee the two will match exactly after any fixed number of rolls.