How this instrument works
The average, or expected value, of a fair die roll is the mean result you would see if you rolled the die an enormous number of times and averaged every outcome. For a die with n equally likely faces numbered 1 through n, that average works out to (n + 1) / 2 — a simple consequence of the faces being a perfectly symmetric run of consecutive integers, whose mean is just the average of the first and last value. A standard six-sided die averages (6+1)/2 = 3.5, sitting exactly between the lowest roll (1) and the highest (6), even though no single roll can ever land on 3.5.
This is a textbook case of expected value, one of probability theory's most useful ideas: it does not predict any one outcome, only the number a large sample of outcomes converges toward. Roll a die once and you get an integer from 1 to n; roll it a thousand times and the running average creeps steadily toward (n+1)/2, a pattern guaranteed by the law of large numbers. The same logic extends to every fair die used in games and gambling — a d4 averages 2.5, a d20 averages 10.5, and a d100 averages 50.5 — and to any other uniformly distributed set of consecutive whole numbers, not just dice.
Knowing a die's average matters beyond curiosity. Tabletop game designers use it to balance damage rolls and turn-order rolls; statisticians use the same (n+1)/2 identity as a worked example of expected value for a discrete uniform distribution; and it is the baseline every dice-based probability question — like the odds of beating a given target number — ultimately compares against.
- Enter the die's face count into Number of sides — it opens on 6, a standard die.
- Read the result from Expected average roll beneath it; it updates the instant the side count changes.
- For unusual dice (d4, d8, d10, d12, d20, d100, or any custom count), just change the side count — the same formula applies to all of them.
- The result is a long-run average, not a possible single roll — for an even number of sides it will always land exactly halfway between two faces.
Worked example — a standard six-sided die
Enter 6 into Number of sides. Expected average roll reads 3.5 — computed as (6+1)/2. That is the number a running average of many rolls converges toward, even though a single die never shows a fraction: over six rolls that happened to land exactly on each face once (1,2,3,4,5,6), the average of those six results is (1+2+3+4+5+6)/6 = 21/6 = 3.5, the same figure the shortcut formula gives directly.
Questions
Why is the average of a six-sided die 3.5, a number it can never roll?
Because the average describes many rolls, not one. A six-sided die's faces are the symmetric run 1,2,3,4,5,6, and the mean of any symmetric run of consecutive integers is exactly the midpoint between the first and last value — here, (1+6)/2 = 3.5. Roll the die enough times and the running average settles closer and closer to that midpoint, even though 3.5 itself never appears on any single roll.
Does this formula work for any number of sides, not just standard dice?
Yes, as long as the faces are consecutive integers starting at 1 — which covers every standard die used in games (d4, d6, d8, d10, d12, d20, d100) as well as any custom die you might specify. The formula (sides+1)/2 does not depend on which specific standard die it is, only on the side count itself.
Why do d20 and d100 average 10.5 and 50.5, both ending in .5?
Because every even-sided die's average lands exactly halfway between two adjacent faces, which always produces a value ending in .5 — a direct result of (n+1)/2 having an odd numerator whenever n is even. An odd-sided die (like a d5) instead averages a whole number, since (n+1) becomes even and divides cleanly by 2.
How is this different from the probability of rolling any specific number?
They answer different questions. The probability of any single specific face on a fair die is always 1/n regardless of which face — every face is equally likely. The average described here is not a probability at all; it is the long-run mean value of the numbers actually rolled, which is why it can land on a number the die itself cannot show.
Does rolling more dice change the average per die?
No — the average of a single die stays (sides+1)/2 no matter how many dice you roll alongside it, since each die is an independent, identically distributed random variable. What changes when you roll multiple dice and sum them is the average of the total, which is simply the number of dice multiplied by one die's average (two six-sided dice, for example, average 3.5+3.5=7 in total, the most common sum in craps).