Due to the sheer number of hypotheses get tested all the time in particle physics, 3 sigma effects appear all the time, and then disappear before getting a larger certainty. That's why it's considered a suggestion, not proof.

And yes, the exact numbers are arbitrary, but their ballpark isn't. You reduce the necessary certainty if you have confidence on your priors, and increase it if you test many different hypotheses. The target confidence also varies from one discipline to another based on how much data one can realistically gather, but nearly all of physics falls on the "we can gather enough data" category anyway.

Well put, but I’ve always felt a middle ground existed where a knowledgeable scientist can make a very educated guess with much lesser statistical significance as long as the experiment is thought through, you are aware of the noise models and alternative explanations have been explored and mostly ruled out. Of course for publication you better whip your ass to whatever sigma they demand, but more often than not there either cause for optimism or pessimism ways before that point.

Not really. The amount of confidence you have grows exponentially with the observed deviation, and that last one grows kinda proportionally to the amount of data you have (if the hypothesis is real, of course).

So there is some fine adjustment possible, but it's very small on the experiments where you need a lot of confidence. There is more margin to judge low non-significant results at the low confidence end, as exponentials grow slowly when they are small.

> Due to the sheer number of hypotheses get tested all the time in particle physics, 3 sigma effects appear all the time, and then disappear before getting a larger certainty.

See, eg, the look-elsewhere effect [1] or the problem of multiple comparisons [2].

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[1] https://en.wikipedia.org/wiki/Look-elsewhere_effect

[2] https://en.wikipedia.org/wiki/Multiple_comparisons_problem

Re: arbitrary. I once attended a Stephen Toulman lecture where he explored the meaning of 'arbitrary.' While the word is often taken to connote "random and uninformed", the root of the word indeed connotes "arbitration," i.e. informed judgement.

In a normal distribution, about 3 out of 1000 values will be over 3 sigma just by random chance.

How many measurements do they make at CERN? They probably see random 3 sigma measurements fairly often.

That's the reason it is expected to have 5 sigma, so that anomalies are extremely rare. The more experiments one makes, the closer one gets to the mean value and this happens exponentially fast. By that, I mean that using some probability theory and some algebra you can get a confidence bound on the accuracy you demand based on the number of experiments that you run.

I think this happens sublinearly w.r.t sample size.

I would like to point you towards Hoeffding's inequality. The probability of deviating more than epsilon from the mean after n experiments is exponential in n.

I think you're talking about different things.

For a fixed range, the probability of being outside that range decays like 1/k^n.

But for a fixed probability, the width of the range with that probability decays like 1/sqrt(n).

If I understand correctly, "certainly new physics" is not correct. It makes the results very substantial, but you still have systemic things to consider at that point, like misapplication of a mathematical concept that no one has caught, or instrument error.

That's correct. I should have said, new physics with high confidence. Physicists tend to be very very conservative when it comes to discoveries as there have been some big blunders during the last century.