ON AVERAGE

Every player does the arithmetic. You look at a unit’s guns, look at the target, and run a quick figure in your head: this does about seven wounds on average, the target has seven, so it dies. The sentence has the shape of a calculation, and it feels like one. It is closer to a guess — and the slack is hiding inside the word “average.”

“Average” is three numbers

When a player says “on average,” they could mean any of three different things, and usually they have not decided which:

• The expected value — the mean. Resolve this attack thousands of times, average the damage, and this is the figure you land on.
• The mode — the single most likely individual result. The tallest bar if you drew the histogram.
• The median — the result you meet or beat exactly half the time. The fifty-fifty line.

For a tidy, symmetric process the three sit on top of each other and the distinction does not matter — roll one D6 and the mean, the median, and the “typical” result are all about 3.5. But the damage an attack deals in 40k is not a tidy, symmetric process, and there the three numbers can pull a long way apart.

Why they pull apart

Resolving an attack is a chain of separate random steps. You roll to hit; each hit rolls to wound; each wound is checked against a save; each unsaved wound then rolls for damage — and that roll is often itself a die, a D3 or a D6. Every stage is a filter, and every filter is a place the attack can come up empty.

That structure gives the damage distribution a particular shape: a heavy lump of probability at zero — every roll in the chain that failed — and then a spread of positive outcomes above it. When the damage per hit is variable, that spread is wide, and a handful of high rolls forms a long tail to the right. A long right tail drags the mean upward, away from the median. That gap is the whole subject of this article.

Example one: when the average is a fair guide

Start with the case where the shorthand works — a high-volume, low-damage weapon fired into a five-strong unit of MEQs (Marine-equivalent models — two wounds apiece, the workhorse mid-durability target).

These are the resolved roll targets the calculation needs: BS is the weapon’s to-hit characteristic, to-wound is the Strength-versus-Toughness result, and save is the target’s armour after the weapon’s AP. Swap in your own weapon’s numbers and everything that follows still holds.

Work out the exact distribution and it is well-behaved. Mean damage 3.3; median 3; mode 3. The three numbers agree. The spread is modest — better than three-quarters of the time the weapon does two to five damage — and the shape is close to symmetric. Convert that to bodies: it kills 1.4 MEQs on average, the most likely result and the median are both one model down, and there is a 43% chance of dropping a second.

Here “on average” is honest shorthand. The mean is also roughly the typical result and roughly the fifty-fifty line, because twenty small, equal dice fold into one tight, near-symmetric pile. A player who says “this clears about a model and a half” is not fooling themselves.

Example two: when it isn’t

Now a different weapon — twelve shots, the same to-hit, to-wound, and save as before, but D6 damage on each unsaved hit — pointed at a single tough model with seven wounds.

The mean damage is exactly 7.0. Seven damage, seven wounds: the heuristic says it dies.

More often than not, it doesn’t. Here are the three numbers:

The probability of actually dealing the seven damage you need is 48% — a hair under a coin flip. And the mode is zero because there is an 11% chance the weapon does nothing whatsoever, every shot turned away at one gate or another: no single positive damage total is more likely than that, since the positive outcomes are spread thin across every value from one into the mid-twenties while every empty result piles onto the same zero.

The player who said “on average 7, so it dies” reached for the mean, attached it to the word “average,” and assumed the mean was also the fifty-fifty line. It is not. The mean sits at 7 because a few big damage rolls haul it up there; the median — the genuine fifty-fifty line — is 6, and the target needs 7. (The Damage Distribution calculator in the Protocol section draws this whole curve; the gap between the mean marker and the median marker is the article in one picture.)

The mean is not the fifty-fifty line

This is the general lesson, and it is worth stating without a weapon attached to it. The question a player actually wants answered is a probability: what are the odds I deal at least the damage I need? A mean that clears the target’s wounds does not make that probability better than even. Whenever the damage distribution leans right — and the filter-chain-plus-variable-damage structure of 40k attacks makes it lean right by default — the mean sits above the median, and the probability of reaching the mean is below half. “On average enough” and “more likely than not” are different claims, and the gap between them is exactly the gap the word “average” hides.

“I should kill this”

Listen to how the heuristic is actually used. “On average I kill this” is rarely a careful statement about a mean — it is a confidence claim in disguise. The player means reliably, or more often than not. “On average” has quietly been promoted to stand in for “I can count on it.” And the on-average number, as the examples have shown, is usually only a roughly-even-odds number. The phrase borrows a confidence the math behind it never issued.

The fix is to stop asking the question forwards. Instead of “what will this weapon do?”, ask it backwards: I want that five-strong MEQ squad gone — how much fire does that take, and how sure do I want to be?

That splits into two steps, and the first is a judgement, not a calculation. Decide the threshold. How sure do you want to be? Eighty percent — likely, with room for the dice to misbehave? Or near-certain, for a play the game turns on? Pick the number deliberately; it is yours to set.

Then do the math. Take a heavier weapon — fewer shots, two damage a hit:

Against a two-wound Marine, two damage is lethal outright: one unsaved wound, one dead model. Clearing a five-strong squad means landing five unsaved wounds, and one shot in six gets through — so the “on average” calculation lands on thirty shots. That is the figure the heuristic hands you.

Thirty shots clears the squad 58% of the time. The “on average” answer is, again, barely better than a coin flip — a long way from the “I should kill this” the phrase was promising.

For an 80% chance you need 39 shots — about a third more than the average implied. For near-certainty, 53 — three-quarters more again. And the cushion is steep on purpose: the whole result rides on just five successful wounds, a small handful of dice, and few dice swing widely. The lighter the volume of fire, the wider the gap between “on average” and “reliably” — which is the subject of the next section. Neither figure is visible in the word “average”; you see them only when you start from the confidence you want and work back to the firepower it costs.

How many dice?

Why did the average hold up in Example one and fail in Example two? Partly the count of dice, partly their size.

The more independent dice fold into a result, the more tightly it clusters around its mean — variance grows more slowly than the total does, so the average becomes a steadily safer guide. Example one is twenty hit rolls feeding twenty wound rolls feeding a handful of saves, every hit worth a flat one damage: many small dice, one tight pile. Example two runs the same kind of chain but narrower, and then each of its two-or-so unsaved hits rolls a D6 — so the result rides on a couple of big, swingy rolls rather than a crowd of small ones.

Push that direction further and you reach the all-or-nothing shot: a single big attack that mostly misses and occasionally deletes its target. Its mean lands in the empty space between the two things that actually happen, describing neither. Few big dice make the average a story about a game you will play exactly once; many small dice make it a forecast you can lean on.

Better questions than “on average”

The shorthand is not banned — it is just underspecified. A few questions sharpen it:

• Ask the threshold question, not the average question. Not “what do I do on average,” but “what is my chance of clearing the wounds I need?” That is the number the decision actually turns on.
• Read three numbers, not one. A plausible floor, the typical result, and the mean. When the floor sits far below the mean, the average is hiding risk, and that is your signal to be careful.
• Let the stakes set the bar. Positive expected value is a fine standard for low-stakes attrition you will repeat all game. For a result that must happen — the shot that has to kill the character — the honest standard is the probability, not the mean: commit enough that the likely outcome clears the bar, not merely the average one.
• More dice when it matters. Splitting a must-kill across more, smaller rolls pulls the distribution in tight around its mean — and makes “on average” trustworthy again.

Closing

“On average” is not wrong. It is underspecified — it compresses a whole distribution into one number and then quietly picks which number that is. The better habit is to keep the distribution in view: not “what happens on average,” but “what do I need to happen, and how likely is it?” The dice will answer the second question honestly. The first one they only seem to.