Why 3d6 feels steadier than a single d20

Two popular ways of generating a number at a role-playing table share the same range and almost the same average. A twenty-sided die produces an integer from 1 to 20, each face equally likely, with an average of 10.5. Three six-sided dice added together produce an integer from 3 to 18, with an average of 10.5 as well. Play both for an evening and they do not feel anything alike.

This note compares the two by counting faces, so every claim can be checked against a table rather than taken on faith.

Counting faces of 3d6

The notation 3d6 means roll three six-sided dice and add them; the free rules write dice the same way, with a modifier after the dice, as in “3d8 + 5” (D&D Beyond — Free Rules: Playing the Game). The total number of ordered combinations is 6 × 6 × 6 = 216. A total of 3 can only happen one way: three ones. A total of 18 can only happen one way: three sixes. A total of 10, by contrast, has 27 ways.

The full count looks like this:

The counts sum to 216, which is the whole point of the table: every ordered roll is accounted for exactly once.

What the table says

Three numbers stand out.

• The middle is crowded. Totals of 10 and 11 together account for 54 out of 216 rolls, or one roll in four. On a d20, a 10 or an 11 accounts for 2 out of 20, or one roll in ten.
• The ends are thin. A 3, a 4, a 17 or an 18 happens on 8 of 216 rolls, under 4%. On a d20, showing a 1, a 2, a 19 or a 20 happens on 4 rolls in 20, a full 20%.
• The shape is symmetric. The count for 3 equals the count for 18, the count for 4 equals the count for 17, and so on, because swapping a die for its opposite face turns any roll into its mirror and the two are equally likely.

That shape, a peak in the middle and thin tails, came out of counting, not assumption. Every ordered roll is equally likely, so the probability of a total is simply its count divided by 216.

Measuring the squeeze

Average alone cannot tell the two methods apart; both average 10.5. Spread can. Standard deviation is the square root of the average squared distance from the mean, and for a flat d20 it works out like this: each face differs from 10.5 by an odd multiple of one half, and squaring those distances and averaging gives:

(1/20) × Σ(x − 10.5)² for x = 1…20 = (1/20) × 665 = 33.25

The square root of 33.25 is about 5.77. For 3d6, the same calculation over the 216 counted rolls gives a variance of 8.75, and its square root is about 2.96 (OpenStax — Mean or Expected Value and Standard Deviation). 3d6 has roughly half the spread of a d20 (2.96 against 5.77), for the same average.

What this does to a character

Now imagine a rule that says a roll must reach 15 or more. On a d20, that succeeds on six faces out of twenty, or 30%. On 3d6, it succeeds on totals of 15, 16, 17 and 18, which is 10 + 6 + 3 + 1 = 20 rolls out of 216, or about 9.3%.

Same average, same broad intent, and a difference of more than three to one (30% against 9.3%) for a character attempting something hard. The reverse holds on the low side: a total of 8 or less happens on 56 of 216 rolls, about 26%, but a d20 result of 8 or less happens 40% of the time.

So a narrow distribution cuts both ways. A character who needs a middling total succeeds reliably, and one who needs an extreme total almost never does. A d20 check swings widely, so a lucky amateur and an unlucky expert both turn up often. A 3d6 check is narrow, so each point of training shifts the chances visibly and an extreme result is rare.

The practical substitution

When moving a rule from a d20 to 3d6, convert by percentage, not by number. Take the chance the d20 rule produced (success on “N or higher”, the form the free rules use), then find the 3d6 total that gives the closest chance. Our table below was built by checking every 3d6 threshold against every d20 threshold.

Read one row as a worked example: a d20 rule that needs a 16 or higher succeeds on 5 faces of 20, 25%. On 3d6 the totals of 13 or higher come up on 21 + 15 + 10 + 6 + 3 + 1 = 56 of 216 rolls, 25.9%. That is the closest match, and a threshold of 14 would drop to 16.2%.

Keeping the same target number, a 16 on 3d6 instead of a 16 on a d20, would cut the chance from 25% to 4.6%. The two distributions diverge most in the tails, and a designer who keeps the number and changes the dice has changed the rules without noticing. Because 3d6 only has 16 possible totals, the match is never exact; each row is the closest available.

A note on modified rolls

Every count above assumes fair dice, each face equally likely. At a table the usual departure from that is not a crooked die but a rule that rerolls, drops the lowest die, or counts a face twice. Each of these reshapes the table above, and each can be read by counting orderings the same way 3d6 was counted. The counting is the method; the die is only the starting point.

Where this stops

Nothing here says one method is correct. Spread is a choice about what a story should feel like: a wide distribution gives luck room to shout, a narrow one makes preparation pay. Naming that choice is more useful than arguing about which die to hand a player, because the math does not care about the argument — it only counts the faces.

Method

All counts come from listing every ordered roll of three six-sided dice (216) and tallying the totals. The d20 variance is the sum of (x − 10.5)² over x = 1 to 20, which is 665, divided by 20. The threshold table compares every 3d6 “or higher” probability with every d20 “or higher” probability and picks the closest. Rules wording and the dice-notation example were read on 2026-10-31 on the page listed below.

Sources

• D&D Beyond — Free Rules: Playing the Game: dice written as a count, d and sides, with a modifier after the dice; the d20 Test succeeds when the total equals or exceeds the target number.
• OpenStax — Mean or Expected Value and Standard Deviation: variance and standard deviation of a discrete distribution.

Written for: a designer who is choosing between one die and a sum of dice and wants the table, not the folklore.