Fire enough shots, and a bullseye paints itself.

The fallacy is named for a Texan marksman who sprays a barn wall with bullets, finds the tightest cluster of holes, and paints a target around it. Inspected after the fact, any spray of randomness contains clusters — the deceit is in pretending the bullseye came first.

A p-value below .05 is a promise about a single, pre-aimed shot: if there were nothing here, results this extreme would arise by luck only one time in twenty. Fire twenty shots — twenty subgroups, twenty outcome measures, twenty foods, twenty trading rules — and "one time in twenty" stops being reassuring and starts being a schedule. The figure above is nothing but that schedule running on schedule.

The trap rarely announces itself, because nobody publishes the nineteen misses. A paper, press release or pitch deck shows you the painted target — left-handed smokers respond to the drug, p = .03 — and the wall full of stray holes stays in a drawer.

Before anyone cheats, the wall is already covered in targets — and that is the sharpshooter's real advantage. Randomness clumps. Ask someone to scatter dots "at random" across a page and they space them out politely, no two too close; genuine randomness does the opposite. When each point lands independently of the last, some pile up and some leave wide gaps. Streaks, clusters and coincidences are not a departure from chance — they are its signature. We simply expect the signature to look tidier than it is, and mistake the clumps for meaning.

The definitive demonstration fell on London. Through 1944 the city was struck by hundreds of V-1 flying bombs, and people were certain they saw a design — some districts hit again and again, others eerily spared, which fed rumours that the untouched streets sheltered German spies. After the war the actuary R. D. Clarke put it to the test. He divided the target area into 576 equal squares and counted the strikes in each: 229 squares hit not once, 211 hit once, 93 twice, 35 three times, 7 four times, and a single square five. The tally tracked a Poisson distribution — the arithmetic of pure, patternless chance — almost square for square. There was no targeting and no spy map. The clusters that unnerved a city were exactly what "random" is supposed to look like.

Psychologists call the misreading the clustering illusion: the reflex to find meaningful groupings in noise. It is why a run of red at roulette feels loaded, why a shuffle that repeats an artist feels broken, and why a handful of cancer cases in one town reads as an outbreak that must have a cause. The sharpshooter barely has to fool you, because the illusion has already done the work — he only needs to hand you a circle for the cluster your eye has helpfully found.

In 2009 the neuroscientist Craig Bennett and colleagues placed a whole Atlantic salmon — purchased at a market, definitively dead — into an fMRI scanner, showed it photographs of people in social situations, and "asked" it to judge their emotions. Then they analysed the scan the way much of the field then did: roughly 130,000 tiny brain regions, each tested separately for task-related activity, with no correction for the number of tests.

A small cluster of voxels in the dead fish's brain cavity lit up as statistically significant. The salmon, by the conventional threshold, was processing human emotion.

The study — which earned an Ig Nobel Prize in 2012 — was a deliberate piece of statistical theatre. At a per-test false-positive rate, 130,000 tests must produce a scatter of spurious hits, and adjacent spurious voxels will sometimes form convincing-looking clusters. Its serious legacy is that multiple-comparison corrections, once skipped in a sizeable fraction of imaging papers, became impossible to omit politely.

The sharpshooter needn't be cynical. As Andrew Gelman and Eric Loken argued, a researcher who runs one analysis can still be implicitly choosing it from a garden of forking paths: exclude that outlier or keep it, adjust for age or don't, take the mean or the median, report endpoint A or endpoint B. Each choice is defensible; together they multiply into dozens of analyses that could have been run — and the one that reaches print is, naturally, one that "worked". No single dishonest step is taken, and the wall still ends up painted.

The known remedies all amount to fixing the target before firing. Pre-register the analysis. Correct the threshold for the number of comparisons — divide it Bonferroni-style, or control the false-discovery rate. Hold out data the choices never touched. And the cheapest remedy of all: treat any surprising, slice-specific finding as a hypothesis for the next study, not a conclusion from this one.

Clinical trials taught the lesson on purpose. The landmark ISIS-2 heart-attack trial (1988) demonstrated convincingly that aspirin saves lives — and its authors, to dramatise the perils of subgroup slicing, also reported that the benefit vanished for patients born under Gemini or Libra. Peter Austin's group later ran the joke in reverse, trawling Ontario hospital records by star sign and duly "finding" zodiac-linked diagnoses. Both were warnings, dressed as findings, about how easily slicing manufactures significance.

Finance backtests. Try a thousand trading rules on the same price history and dozens will be wildly "profitable" in-sample, for exactly the reason the dead salmon thought about photographs. The strategies that get sold are the painted targets; the out-of-sample future is the part of the wall nobody shot yet.

Cancer clusters and headline nutrition. The term "Texas sharpshooter" comes from epidemiology, where apparent disease clusters around some landmark must be weighed against the thousands of neighbourhoods in which nobody went looking. Food-and-health headlines run the same engine: survey hundreds of foods against dozens of outcomes and chance alone keeps the front pages stocked.

The protective question never changes: how many shots were fired before this target was painted? Ask how many subgroups, endpoints, foods or strategies were tested; whether the hypothesis predates the data; whether the threshold was corrected; whether it replicates on a wall the shooter hasn't seen. A genuine bullseye survives all four questions. A painted one rarely survives the first.