When you can use a normal approximation — and when it'll lie to you.

Where it works: sums of fair dice

The central limit theorem says: if you sum a large number of independent random variables with finite variance, the distribution of the sum, suitably standardised, approaches a normal (Gaussian) distribution. "Suitably standardised" means subtracting the mean and dividing by the standard deviation, so you end up looking at how many SDs from the mean a given outcome sits.

Visually, this is what's happening at every step from 1d6 to 2d6 to 4d6:

1d6 is flat. 2d6 is a tent. 4d6 is a clearly mound-shaped histogram. 8d6 is, to the eye, a bell curve. By the time you're summing a fistful of dice, the normal approximation is good enough for most practical purposes — within a percentage point or two on most queries.

With mean μ and standard deviation σ in hand, the normal approximation gives you a quick rule of thumb:

• About 68% of outcomes within ±1σ of μ.
• About 95% within ±2σ.
• About 99.7% within ±3σ.

For 8d6: μ = 28, σ = √(70/3) ≈ 4.83. So roughly 68% of rolls land in [23, 33], 95% in [18, 38]. Compare to the panel: that's close to what the bars actually say. Not identical — but close.

Where it fails, and how loudly

Four common dice situations where the normal approximation gives you the wrong answer in ways that actually matter:

1. Few dice

1d20+5 is a single die plus a constant: a uniform distribution shifted right. Approximating a uniform distribution as a normal is visibly wrong — the tails of a normal stretch to infinity, while a uniform truncates abruptly. The central limit theorem needs enough dice for the bell shape to emerge; one die is not enough, and even 2d6 is still tent-shaped, not bell-shaped.

2. Heavy-tailed dice (exploding, fat-tail mechanics)

Exploding dice (1d6! — on a 6, roll again and add) have a long right tail with non-zero probability extending far past the original face count. The normal approximation systematically under-estimates the probability of those far-right outcomes, because the bell curve falls off too fast compared to a fat tail.

See exploding dice and fat tails for the worked-out version. Short version: don't normal-approximate exploding dice.

3. Asymmetric mechanics (advantage, disadvantage, kept-best-of)

2d20kh1 (advantage) is the maximum of two 1d20 rolls — an order statistic, not a sum. Order statistics are skewed, especially at the extremes. The distribution of kh1 bunches toward the high end; kl1 bunches toward the low end. Neither looks symmetric. A normal approximation, which is symmetric by construction, smooths over the skew that's the whole point of the mechanic.

4. Small mean, hard floor

A roll like 2d4 has mean 5 and SD ≈ 1.58. The normal approximation says ~16% of outcomes are below 3.4 (one SD down), but the actual minimum is 2 — and the distribution sits in a very narrow band. When μ − 3σ < min, you've got "the bell curve says probability extends below where the dice physically can land", which is always wrong by exactly whatever probability the bell curve assigns to the impossible region.

The rule of thumb

Use the normal approximation as a sanity check, not as the answer. For the four-or-more-dice flat sum case it's accurate to within a percentage point, which is fine for napkin work. For everything else the engine on this site computes the exact probability in microseconds — there is no reason to approximate.

A pragmatic test: compare the closed-form normal answer to the panel for the same expression on this site. If they agree within ~1% across the range you care about, the approximation is fine. If they don't, trust the engine — the rationals can't be wrong because they're not rounded.

Try it yourself

↦ /strike/4d6 — bell-ish ↦ /strike/8d6 — bell-yes ↦ /strike/1d20 — uniform, normal would lie ↦ /strike/1d6! — fat tail, normal would lie ↦ /strike/2d20kh1 — advantage, skewed, normal would lie