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Motivated by this question in Statistics community: Why do physicists use sigma while biologists use p values/posterior probabilities?. After discussion it essentially reduces to: why assuming normal distribution works so well in the particle physics, that one can speak of "sigmas" instead of proper p-values? Naïvely, this is the consequence of the central limit theorem, applied to measuring the result of a large number of (collision) events. However, again naïvely, we are talking here about a binomial rather than a normal distribution (one even per billions taking place, rather than adding contributions from the billions of events.)

I will appreciate a discussion by experts in particle physics.

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  • $\begingroup$ If to look in general,- binomial is discrete distribution, while many Physical variables are non-countable and continuous. Even in particle physics you only get such impression that it's countable due to limited set of experiments and/or events with energy ranges. If you would expand all things to a greater scale,- normal distribution would be seen clearly as in most Physics. It happens probably because most measurements have some top expected value under certain conditions. And this likely follows from determinism and causality principle. $\endgroup$ May 15 at 21:20

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However, again naïvely, we are talking here about a binomial rather than a normal distribution (one even per billions taking place, rather than adding contributions from the billions of events.)

This 'one event per billions' relates to the type I error rate. It is not the distribution of the signal.

The signal is often a sum of many little variables which is why it becomes approximately normal distributed.

For example, in the case of high energy physics we often deal with mass spectra and counts which are Poisson distributed. The Poisson distribution becomes approximately a normal distribution when you increase the observation window (which is like adding many individual Poisson distributions together).

Example illustration

$$X_i \sim Poisson(100) \\ |X_i-100|>30 \sim Bernoulli(0.00237) $$

example of different distributions

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The number of sigmas $z$ (significance) is equivalent to the p-value, in fact the relation is a 1-to-1 function: $$z = \Phi^{-1}(1-\text{p-value}).$$ Where $\Phi$ is the CDF of the standard normal distribution.

It is just a more practical way to quote small p-values.

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    $\begingroup$ What is $\Phi$ here? Just a constant? $\endgroup$ May 15 at 22:18
  • $\begingroup$ This is true only for normal distribution, and the question is about why/where the normal is justified in particle physics. $\endgroup$
    – Roger V.
    May 16 at 4:38
  • $\begingroup$ @RogerV. it is just a definition, there is no true or false here. It can be not convenient, misleading, ... in some cases, but it is just a definition. $\endgroup$ May 17 at 7:44

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