Understanding the average roll of dice
The "average" of a die is its expected value — the mean result you would get if you rolled it a very large number of times and averaged all the outcomes. For a fair die whose faces are the consecutive whole numbers 1 through S, every face is equally likely, so the average is simply the midpoint of the range. That gives the classic formula:
- One fair die (faces 1…S): average = (S + 1) / 2.
- D dice, each with S sides: average total = D × (S + 1) / 2.
The reason the two-dice formula is just "add the averages" is that expected values add regardless of whether the dice are independent. You never have to enumerate every combination — the average of a sum is the sum of the averages.
Why it is (S + 1) / 2
The numbers 1 through S form an arithmetic sequence. Their sum is S(S + 1) / 2 (Gauss's formula), and dividing by the S equally likely faces cancels the S, leaving (S + 1) / 2. For a standard six-sided die that is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5. Notice 3.5 is exactly halfway between 1 and 6, which is what the midpoint interpretation predicts.
Common reference points
- d4: average 2.5 — a d4 pair (2d4) averages 5.
- d6: average 3.5 — 2d6 averages 7, 3d6 averages 10.5, 4d6 averages 14.
- d8: average 4.5. d10: average 5.5. d12: average 6.5.
- d20: average 10.5 — the single most-cited value in tabletop games.
- d100 (percentile): average 50.5.
Averages, minimums, and maximums
Because faces run from 1 to S, D dice can total anywhere from D (all ones) to D × S (all sixes on d6s). The average always sits exactly in the center of that range: for 3d6 the total spans 3 to 18, and the average 10.5 is the midpoint. The distribution of a single die is flat (uniform), but as you add dice the totals bunch toward the middle in a bell-like shape, so the average also becomes the most likely region of outcomes.