What sigma actually measures
Sigma (σ) is the standard deviation: a unit of how spread out a set of measurements is. Saying a result is "three sigma" means it sits three standard deviations away from what you would expect if nothing interesting were happening. The further out it sits, the harder it is to explain as a fluke of random noise. The number itself is not a probability — it becomes one once you ask how much of the normal distribution lies beyond that point, which is exactly what the p-value is.
P-values, and what they do not mean
The p-value is the probability of seeing a result at least as extreme as yours assuming there is no real effect. A p of 0.05 means that if nothing were going on, you would still get a result this striking about one time in twenty. What it does not mean is that there is a 95% chance your hypothesis is correct — that is the single most common misreading in all of statistics. The p-value describes the data given a hypothesis, not the hypothesis given the data.
One tail or two?
Use two tails when you are asking whether something is different, in either direction. Use one tail when only one direction would count as an effect. Two-tailed p-values are exactly double their one-tailed counterparts, so the same sigma looks less impressive under a two-tailed test. Particle physics conventionally quotes one-tailed values, which is worth remembering when you compare a physics result against a clinical trial.
Why physics waits for five sigma
Most fields treat p < 0.05 (roughly 2σ) as publishable. Particle physics does not: it calls 3σ "evidence" and reserves "discovery" for 5σ, a p-value near 3 in 10 million. The reason is the sheer number of comparisons. When you scan millions of collision events across many energy ranges, rare coincidences become routine, and a 2σ bump somewhere is almost guaranteed. The famously high bar is not physicists being fussy — it is the correct response to looking in a very large number of places at once.
Frequently asked questions
What does 5 sigma mean?
It means a result sits five standard deviations from the no-effect expectation, with a one-tailed p-value of about 1 in 3.5 million. In particle physics it is the conventional threshold for claiming a discovery rather than mere evidence.
How do you convert sigma to a p-value?
The p-value is the area of the normal distribution beyond that many standard deviations. One-tailed, p = 1 − Φ(σ); two-tailed, p = 2(1 − Φ(σ)), where Φ is the cumulative normal distribution. This calculator does it in both directions.
Is a p-value the probability that my hypothesis is wrong?
No. It is the probability of data this extreme if the null hypothesis were true. Turning it into a probability about the hypothesis itself requires Bayesian reasoning and a prior, which the p-value does not contain.
What p-value corresponds to 3 sigma?
About 0.00135 one-tailed, or 0.0027 two-tailed — roughly 1 chance in 740, or 1 in 370 respectively. Physics calls this level "evidence", not a discovery.
Why is 0.05 the usual significance threshold?
Largely convention. Ronald Fisher suggested it as a convenient rule of thumb in the 1920s and it stuck. There is nothing fundamental about it, and many fields now argue it is far too lenient for the number of hypotheses actually being tested.
References
- Particle Data Group, Review of Particle Physics — statistics chapter, on the 3σ/5σ conventions.
- Lyons, L. (2013). Discovering the Significance of 5 sigma. arXiv:1310.1284.
- Wasserstein, R. & Lazar, N. (2016). The ASA Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2).