Two numbers define a crit
A critical hit system usually has two parameters: the chance that an attack crits, and the multiplier applied when it does. Say an attack deals 100 damage normally, crits 20 percent of the time, and does double damage on a crit.
Each attack then has two possible outcomes. With probability 0.8 it deals 100, and with probability 0.2 it deals 200. Those two numbers and their probabilities are everything needed to describe the damage.
Average damage
The average, or expected damage, weights each outcome by how likely it is:
0.8 times 100, plus 0.2 times 200, equals 80 plus 40, which is 120.
A shorter way to get the same figure is to write the average as base damage times (1 plus crit chance times the bonus part of the multiplier). Here the bonus part is the multiplier minus 1, which is 1, so the average is 100 times (1 + 0.2 × 1) = 120. In other words, a 20 percent chance of doubling adds 20 percent to the average.
This is why crit chance and crit multiplier are interchangeable at the level of the average. What matters is their product: chance times the bonus part. 20 percent at 2x and 40 percent at 1.5x both have a chance-times-bonus value of 0.2 (0.2 × 1 and 0.4 × 0.5), so both average 120 per attack. But the experience of using them is different, and the next sections explain why.
Spread: why the same average can feel different
Variance measures how far outcomes typically sit from the average. For one attack in the example above, the deviations from the average of 120 are 20 below (for a normal hit) and 80 above (for a crit). Squaring and weighting them:
0.8 × 20² + 0.2 × 80² = 0.8 × 400 + 0.2 × 6400 = 320 + 1280 = 1600.
The standard deviation is the square root of 1600, which is 40. So a typical attack lands about 40 away from the average of 120, which is large compared with the average itself.
Now compare two builds that both average 125 per attack:
| Build | Normal hit | Crit hit | Crit chance | Std. dev. per attack |
|---|---|---|---|---|
| Rare, big crits | 100 | 200 (2x) | 25% | about 43 |
| Frequent, small crits | 100 | 150 (1.5x) | 50% | 25 |
Checking the first row: 0.75 × 25² + 0.25 × 75² = 468.75 + 1406.25 = 1875, and the square root is about 43.3. The second row: both outcomes sit exactly 25 from the average of 125, so the standard deviation is 25.
The averages match, yet the first build is far more erratic. Players often describe the second as “reliable” and the first as “swingy”. Neither is better by the average alone; the choice depends on whether the situation rewards consistency or occasional spikes.
Variance shrinks over many attacks
Individual attacks are noisy, but totals behave more predictably. If attacks are independent, the standard deviation of the total grows with the square root of the number of attacks, while the total average grows in direct proportion to it.
For the 20 percent, double-damage example, the average is 120 per attack and the standard deviation is 40:
- 10 attacks: average 1200, standard deviation 40 × √10, about 126. That is about 10.5 percent of the average.
- 100 attacks: average 12,000, standard deviation 40 × √100 = 400. That is about 3.3 percent of the average.
So across a hundred attacks, the total is relatively close to what the math says. Across ten, a fight can come in noticeably above or below it. This is the basic reason that long fights reward the stronger build more consistently, and short bursts of damage are where crit luck matters most. The same principle underlies why outcome certainty improves with repeated trials.
Streaks of crits are normal
A run of three crits in a row at 20 percent has a probability of 0.2³, which is 0.008, or 0.8 percent, for any particular starting attack. That sounds rare. But over 100 attacks there are 98 places such a run could begin, and 98 × 0.008 is just under 0.8, so seeing one or more such runs somewhere is entirely ordinary. Players who feel their crits “came in a clump” are usually describing a pattern that random sequences produce regularly.
Where crits interact with other damage rules
Crit systems rarely work alone. A few interactions are worth keeping in mind:
- Additive versus multiplicative stacking. Several crit bonuses may add together before being applied, or multiply. The average damage formula changes accordingly, and multiplicative stacking raises the top end faster.
- Damage scaling. Where damage is reduced by armor or scaled by combo position, a crit multiplies a reduced base, and the practical gain differs from the headline number. The ideas behind damage scaling show up in many fighting games and RPGs.
- Time to kill. Crits matter most when they decide whether a target dies in fewer hits. Because of thresholds, a build with a lower average can still win a fight more often if its spikes cross a key breakpoint. The relationship between crit variance and time to kill is a design topic of its own.
- Guaranteed or capped crits. Some games make crits deterministic in some conditions, which removes variance entirely and leaves only the average.
What to take from the numbers
Average damage is the expected value of an attack, and it tells you what a build does over a long stretch. It also explains why the sample size matters when judging a build from a handful of fights. Standard deviation tells you how uneven the stretch will feel. When comparing builds, it helps to look at both: the same average with a smaller spread suits situations where a fight is short or where one failure is costly, and a larger spread suits cases where only an unusually high result will do.
None of this predicts any given attack. It describes the range of results a player should expect to see, so that an unusually good or bad sequence can be recognised as part of the normal range rather than as a sign that something has changed.