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In particle physics experiments, one often quotes the result of measurement of an observable with $1\sigma$, $2\sigma$, $3\sigma$ ranges. The experiments typically give a best-fit value with a $3\sigma$ range for that observable. Let us take an example. The best-fit value of the neutrino solar mixing angle $\theta_{12}$ is $33.62°$ with a $3\sigma$ range $31.42°$-$36.05°$. To get some ideas I was looking at the links How Many Sigma? and What does a 1-sigma, a 3-sigma or a 5-sigma detection mean?. But I failed to understand several points.

  • Why is the range $\theta_{12}\in[31.42°,36.05°]$ called a $3\sigma$ range? Does it mean that the probability that $\theta_{12}$ lies between $31.42°$ and $36.05°$ is $0.9973$. Equivalently, the chance of getting a value between $31.42°$ and $36.05°$ is $99.73\%$?

  • When one talks about $\sigma$'s, they must generate a Gaussian or Normal distribution with a mean $\mu$ and a standard deviation $\sigma$. How is this Gaussian distribution obtained? What is the best fit value has to do with this distribution?

  • What do we mean when we say that some value is disfavoured at $3\sigma$? Of course, we mean that it lies outside the $3\sigma$ range. However, in terms of probability, is it that if a value is outside the $3\sigma$ range, we mean, that probability of obtaining it is $\leq 0.27\%$?

  • You have a normal distribution you can always calculate any $N\sigma$ range. How is it possible that the $3\sigma$-range is known but not the $5\sigma$?

  • Can we expect that $3\sigma$ range of an observable to become narrower in future?

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I am a bit rusty on my statistics so the following may not be the most precise.

Why is the range $\theta_{12}\in[31.42°,36.05°]$ called a $3σ$ range? Does it mean that the probability that $\theta_{12}$ lies between $31.42^o$ and $36.05^o$ is 0.9973?

Yes. I have more commonly seen either a z (or t)-score $z=\dfrac{x-\mu}{\sigma}$ or an $x\%$ (in this case 99.73%) confidence interval, but the idea is the same.

When one talks about $\sigma$'s, they must generate a Gaussian or Normal distribution with a mean $\mu$ and a standard deviation $\sigma$.

No, not necessarily. The standard deviation $\sigma$ simply refers to the quantity $\sigma = \sqrt{\langle x^2\rangle-\langle x\rangle^2}$. However, when creating confidence intervals or more general hypothesis testing one wants the sampling distribution as close to normal as possible as this gives meaningful interpretations. This distribution is naturally obtained. (the number of samples being greater than $30$ is one criteria for test validity since distributions tend towards normal; see Central Limit Theorem)

...in terms of probability, is it that if a value is outside the $3\sigma$ range, we mean, that probability of obtaining it is $\leq 0.27%$?

Yes. The reason we want (some high number)$\sigma$ is that we can never statistically prove anything (i.e. with 100% confidence), so in general we assume that some event happens due to random chance (null hypothesis), and if that chance is $<x\%$ then it is statistically significant and we may reasonably assume that it is not mere chance; that some other factor may explain the phenomena.

How is it possible that the $3\sigma$-range is known but not the $5\sigma$?

I do not know of a case in which this is true. In general, one may always construct any $\sigma$-range; however, if not many samples are taken it is more useful to take lower $\sigma$ values as the interval's spread may be too large at higher values for any meaningful analysis.

Can we expect that $3\sigma$ range of an observable to become narrower in future?

Generally, yes. If one takes more samples the range (at a fixed $\sigma$ value) should become smaller, proportional to $n^{-1/2}$

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