What Does 5 Sigma Actually Mean?

On 4 July 2012, CERN announced the Higgs boson at five sigma, and a room full of physicists applauded a number most people in the wider audience could not define. Five sigma is the rule that decides when a measurement becomes a discovery. It is also badly misunderstood, routinely reported as “99.9999 percent certain the particle is real,” which is not what it says. This is what a sigma actually measures, why the bar sits at five, and why two honest statisticians can look at the same data and disagree about whether anything has been found.

What a p-value is, and the thing it is not

Start with the quantity underneath the sigma, which is the p-value. Suppose there is no new particle, no signal, only the familiar background. The p-value is the probability that random fluctuations in that background would, by chance alone, produce a bump at least as large as the one you saw.

Small p-value, rare fluke, so the background-only story looks strained. That is the whole logic. A sigma count is just this probability rewritten as a distance, in standard deviations, out along the tail of a bell curve. One sigma is a p-value around 0.16, hardly rare. Three sigma is about one in 740. Five sigma is roughly one in 3.5 million.

Now the part almost every headline gets wrong. The p-value assumes the null hypothesis is true and asks how surprising the data would be. It does not tell you the probability that the null hypothesis is true. Five sigma is not a 99.9999 percent chance the Higgs is real. It is a statement that if the Higgs were not there, data this striking would appear about once in 3.5 million tries. The direction of the conditional matters, and reversing it is the single commonest error in science reporting.

Why the bar is five, and not three

Five sigma is a convention, not a law of nature. Particle physics keeps two bars: three sigma to call something “evidence,” five to call it a “discovery.” Most journals will not let the word discovery near a title below five. The statistician Louis Lyons, who has written the standard defences of the threshold, gives three reasons it sits so high.

History. Too many three and four sigma signals have simply evaporated once more data arrived. The graveyard is large enough that the field learned to distrust anything short of five.

The look-elsewhere effect. Scan a wide enough range and a fluctuation somewhere becomes almost guaranteed, even when nothing real is present. Lyons’s colleague Glen Cowan makes the point with a plot of pure random noise in a hundred bins: the eye picks out several convincing “peaks,” and every one is empty. The p-value for one specific spot overstates things. The honest number corrects for all the places you could have looked, and that correction costs you significance.

Systematics. A sigma is the signal divided by the uncertainty in the background. If you have underestimated that uncertainty, because some instrumental effect is mis-modelled, a three sigma bump can be pure artefact. Demanding five buys a margin against the systematic errors you have not thought of.

Lyons himself is not dogmatic about it. He has argued that a fixed bar is crude. An extraordinary claim, such as neutrinos outpacing light, should demand far more than five, while a well-predicted one like the Higgs might reasonably need less. The rule endures anyway, partly for its simplicity and partly for fairness. A shared, stated-in-advance threshold stops credit going to whichever team is willing to publish on the flimsiest evidence.

The sigma scale at a glance

Here is the whole scale in one picture, with four results from this article marked on it. Watch the dot furthest to the right. OPERA cleared the discovery bar comfortably, and it was still wrong.

The sigma scale, from pure chance to discovery

Six sigma, and wrong: the OPERA cable

In September 2011 the OPERA experiment reported neutrinos arriving from CERN at Gran Sasso about 60.7 nanoseconds early, which is to say faster than light. After a repeat run the significance reached 6.2 sigma, past the discovery bar, a result that would have overturned special relativity.

It was wrong. A fibre-optic connector carrying a GPS timing signal to the master clock was not fully screwed in, adding a delay of around 73 nanoseconds. A clock oscillator running fast pulled about 15 nanoseconds the other way. The two corrections nearly cancelled and left roughly 58 nanoseconds, almost exactly the anomaly. The neutrinos had obeyed the speed limit all along.

The lesson is the one the five sigma rule cannot fully enforce. OPERA’s statistics were immaculate. The 6.2 sigma described the precision of the measurement, not its accuracy, and a loose cable lives in the gap between the two. No p-value protects against an apparatus that is quietly lying to you.

The anomaly that vanished without new data

For twenty years the muon’s magnetism was the most watched crack in the Standard Model. Measured, it came out slightly larger than theory predicted, and the gap refused to close. Against the 2020 theory prediction, Fermilab’s 2021 measurement put the discrepancy at 4.2 sigma, tantalisingly near discovery.

Then the gap dissolved, and almost none of the movement came from the experiment. The hard part of the prediction is the contribution of the strong force. It can be computed two ways: from older collider data, or from lattice simulations of the theory on a grid. The two disagreed. When the 2025 theory update adopted the lattice value, the predicted number rose to meet the measurement, and the discrepancy fell to statistical noise. Fermilab’s own final result, more precise than ever, agrees with the revised theory.

The other question: how much do you now believe?

A p-value is deliberately one-sided. It scores how badly the no-signal story fits, and stays silent on how much you should believe the alternative. Many physicists, especially in cosmology, want that second question answered directly, and for it they turn to Bayesian model comparison.

The tool is the Bayes factor: given the data, how much have the odds between two models shifted? Its signature feature is a built-in penalty for complexity. A model with a spare free parameter can accommodate many possible datasets, and that flexibility counts against it. To win, the richer model must fit not just better but enough better to justify the freedom it claimed. Add a parameter that the data barely constrain and the Bayes factor drifts back toward even odds, its verdict being that the data did not really move you.

Cosmologists read the result on the Jeffreys scale, stated by Roberto Trotta. Odds around 3 to 1 count as weak, 12 to 1 moderate, 150 to 1 strong. Because the scale runs on the logarithm of the odds, climbing one rung takes roughly ten times more supporting data. Evidence, on this view, accrues slowly and grudgingly.

When the two frameworks disagree, on purpose

Because the two approaches ask different questions, they can return answers that look contradictory while both are right. The clearest live example is dark energy.

After a 2026 recalibration, the case for dark energy evolving over cosmic time stood at 3.2 sigma by the frequentist count. The same dataset, weighed as a Bayes factor, gave odds of only about 5 to 1, which the Jeffreys scale files under weak. One number sounds like a near-discovery; the other sounds like a shrug. Neither is wrong. The frequentist figure asks how rare the data would be with no evolution, and the answer is fairly rare. The Bayesian figure asks whether adding evolution, with its extra parameters, was worth it, and the answer is not especially. We walk through the physics in our piece on what dark energy is.

Each framework has a matching weakness, and honesty means owning both. The frequentist p-value is hostage to the look-elsewhere effect and to how well you modelled your systematics. The Bayesian evidence depends on the prior you place on the extra parameter, and a defensible change of prior can shift the verdict by a full rung on the scale. There is no framework that removes judgement. There are only two ways of making the judgement explicit.

A p-value done right

None of this makes the machinery worthless. It makes it a tool with a domain, and there are places it works almost perfectly.

A Bell test is the cleanest case. The null hypothesis is sharp: the world runs on local hidden variables. There is one number to compute, no free parameters to tune, no wide spectrum to scan for a bump. When a 2023 superconducting experiment violated the Bell bound, it reported a p-value below ten to the power of minus 108. That is a frequentist statement with no look-elsewhere ambiguity and no systematics quietly inflating it, which is exactly why it carries such weight. The framework is not the problem. Using it outside its domain, and misreading what it says, is.

How to read a significance claim

A few habits separate a durable claim from a fragile one.

Ask what the null was. A p-value is only as meaningful as the hypothesis it tries to reject, and a sharp null beats a vague one. Ask whether the number is local or global. A raw local p-value that ignores the look-elsewhere effect is optimistic by design. Ask where the uncertainty is. If a significance rests on a theory prediction as much as a measurement, the prediction can move, as the muon showed. Ask which framework is talking. A frequentist sigma and a Bayesian odds are answering different questions, and quoting one while imagining the other is how a weak result gets dressed as a strong one.

Established, contested, unproven

Established. The definitions. A p-value is a tail probability under the null, sigma is that probability as a distance, and five sigma is about one in 3.5 million. The two-tier convention, evidence at three and discovery at five, is standard across particle physics. The look-elsewhere effect and systematic uncertainty are real and quantifiable.

Contested. Whether five sigma is the right bar. Lyons and others argue a claim-dependent threshold would serve better, stricter for the implausible and looser for the well-motivated. More broadly, whether frequentist or Bayesian methods are the sounder default is a decades-old argument statisticians have not settled. To a real extent it is a question about what you want a number to mean.

Unproven, by construction. No amount of sigma proves a discovery is real. The figure is always conditional on a model of the background and the apparatus, and OPERA is the standing reminder that a flawless statistical significance can sit on top of a loose cable. The number is a guide to how seriously to take a result. It is never the last word.