The Texas Sharpshooter Fallacy: How a Statistical Error Can Send Someone to Prison

What the Texas Sharpshooter Fallacy Actually Means

The name comes from a joke about a Texan who fires at a barn wall, then walks over and draws a target around whichever cluster of bullet holes is densest. He shows off his grouping and declares himself a marksman. The cluster is real. The accuracy is fabricated.

In statistical reasoning, the fallacy describes any process by which data are selected or grouped after the fact to match a hypothesis that was formed in response to the data. It is a form of circular reasoning. The apparent pattern is genuine only within the cherry-picked subset; it dissolves when the full dataset is examined. And because the subset looks, superficially, like evidence, the fallacy is genuinely difficult to detect without examining what was excluded.

In criminal trials the fallacy is particularly dangerous because the presentation is authoritative, the data are real, and the jury has no independent means of checking what was left out. A chart, a table, a graph. Each carries the weight of objective evidence. None of them automatically discloses its own denominator.

How Shift-Attendance Charts Become Misleading Evidence

A shift-rota attendance chart used in a criminal trial works as follows. Investigators select a set of suspicious events (collapses, deaths, rapid deteriorations) and then cross-reference them against the nursing rotas to identify who was on duty for each. If one nurse appears consistently across the list, this pattern is offered to the jury as evidence of her culpability.

The appearance of proof is created by the intersection of two things: the list of events and the attendance record. But the evidential weight of that intersection depends entirely on how the list of events was constructed. If events were included in the list partly on the basis that the suspect nurse was on duty, because those are the events investigators focused on, the ones clinicians remembered in the context of the suspect, the ones that survived the selection process because they aligned with the emerging hypothesis, then the attendance pattern in the chart does not confirm the hypothesis. It simply reflects the selection that created it.

This is precisely the critique applied to the prosecution's shift-rota chart by statisticians who have examined the prosecution shift-rota chart and the selection-bias critique in detail. The chart showed one nurse present at every one of twenty-five suspicious events. But the twenty-five events were not selected from a complete and independent list of all deteriorations on the unit during the period. They were selected, at least in part, through a process in which her presence was a factor in determining which events were treated as suspicious.

The Denominator Problem: What the Chart Did Not Show

Every statistical argument about patterns of presence has two components: a numerator and a denominator. The numerator is the number of events associated with the suspect. The denominator is the total population of events against which those associations should be measured.

A chart that shows the suspect nurse present at twenty-five suspicious events, without also showing how many non-suspicious events she was not present at, how many other nurses were present at the same events, and how many deteriorations occurred when she was not on duty, is missing its denominator. It is not a complete picture. It is a numerator presented as if it were a ratio.

Hours Worked Versus Events: A Fundamental Adjustment

One specific denominator problem in shift-attendance evidence concerns the base rate of hours worked. A nurse who works an above-average number of unsociable shifts (nights, weekends, bank holidays) will naturally appear in a higher proportion of crisis events simply because more crises happen to fall within her extended shift presence. Any analysis that fails to adjust for total hours worked will overstate the association between her presence and adverse outcomes.

The basic question any competent statistical analysis must answer is not whether the nurse was present at every suspicious event, but whether her rate of association with suspicious events exceeds what would be expected given her overall rate of presence on the unit. That adjustment is not exotic statistics. It is elementary. Where it is not performed, the chart is structurally misleading regardless of whether its raw data are accurate.

Why Professional Statisticians Raised the Alarm

Several senior professional statisticians have commented publicly on the statistical evidence in the Letby case, and they share a common concern: the standard analytical safeguards were not applied.

Professor Richard Gill of Leiden University, who was a central figure in identifying the statistical errors that led to the wrongful conviction of Dutch nurse Lucia de Berk, has characterised the chart shown to the jury as a textbook application of the Texas sharpshooter fallacy. Professor Norman Fenton has conducted a formal Bayesian analysis of the available evidence and concluded the posterior probability of guilt does not meet the criminal standard when natural-cause base rates are properly incorporated. The Royal Statistical Society has issued public statements noting its concern with selection-effect problems in attendance-based evidence presented to criminal juries.

None of these experts is making a claim about guilt or innocence. They are making a claim about methodology: the statistical argument presented to the jury does not meet the standards the profession sets for valid statistical inference in a legal context.

Historical Precedents: Sally Clark, Lucia de Berk

The history of wrongful convictions involving misapplied statistics in cases of infant and patient death is well documented.

Sally Clark, a British solicitor, was convicted in 1999 of murdering her two infant sons partly on the basis of statistical evidence from paediatrician Roy Meadow, who estimated the probability of two sudden infant deaths in one family as 1 in 73 million. The Royal Statistical Society intervened publicly to explain that this calculation was methodologically wrong on multiple grounds, including the failure to account for genetic and environmental correlations. Clark's conviction was quashed on appeal in 2003 after a separate evidentiary problem emerged, but the statistical critique was foundational to the case's reassessment.

Lucia de Berk, a Dutch nurse, was convicted in 2003 of multiple murders and attempted murders on a neonatal unit, substantially on the basis of a statistical calculation suggesting her association with suspicious events was too improbable to be coincidental. The calculation was subsequently shown to be wrong, for many of the same structural reasons now being applied to the Letby shift-rota chart. The Dutch Supreme Court acquitted her in 2010. The statistical work that drove her exoneration was led in part by Professor Gill.

What Courts Should Require Before Admitting Attendance Evidence

The post-Clark guidance from the Royal Statistical Society establishes clear standards for statistical evidence in UK criminal trials: the statistical methodology must be disclosed and independently reviewable; confidence intervals and alternative hypotheses must be presented; the selecting process for the data underlying any chart must be documented; and, where possible, an independent statistician should review the methodology before it is presented to a jury.

These standards are not uniformly applied. Their absence does not automatically make a verdict unsafe. But it does mean the jury has been asked to weigh statistical reasoning it cannot verify, using a methodology the defence may not have had adequate tools to challenge. When professional statisticians subsequently review that methodology and conclude it is structurally flawed, the evidential foundation for the attendance-based argument is materially undermined.