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In statistics, $\sigma$ or $1\sigma$ is the standard deviation. For the normal distribution, approximately $68\%$ of the values lie within $1\sigma$ range, $95\%$ within $2\sigma$, and $99.7\%$ within $3\sigma$. These are the things I know.

However, as I read papers, I came across two situations, which are:

  1. Massara2021

The errors on the neutrino masses from the marked power spectrum are $\sigma_{cb}(M_\nu)=0.5 \ \text{eV}$ and $\sigma_{m}(M_\nu)=0.017 \ \text{eV}$. The latter indicates that marked power spectra on the total matter density can put a $3.5\sigma$ constraint on the minimum sum of the neutrino masses.

I know that the smaller $\sigma$ is, the tighter the constraints on the parameters. But, what does $3.5\sigma$ mean? How is it calculated to?

  1. Houliston2022

In the joint analysis, the Hubble constant found in flat $\Lambda$CDM is $H_0 = 73.3^{+1.7}_{-1.8}\text{km}\cdot \text{s}^{−1} \text{Mpc}^{−1}$. This $2.4\%$ precision value agrees with the SH0Es 2020 value, and is in $3.1\sigma$ tension with the Planck CMB value.

What does $3.1\sigma$ tension mean?

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  • $\begingroup$ The exact relation is $P(\le \! k\sigma) = \mathrm{erf}(k/\sqrt{2})$, where $\mathrm{erf}(x) \equiv \frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}}\,dt$ is the error function. Thus, $P(\le \!1\sigma) = 0.683$, and $P(\le \!3.1\sigma) = 0.998$. $\endgroup$
    – pela
    Commented Jul 5 at 12:15
  • $\begingroup$ @pela I understand these relations, but I do not understand k=3.1 comes from. $\endgroup$
    – Wang Yun
    Commented Jul 5 at 12:36
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    $\begingroup$ Saying "the tension is at 3.1σ" is another way of saying "we're 99.8% sure both results cannot be true". The number is calculated from the known random and systematic uncertainties, as seen in Lenard Kasselmann's answer. But is your question perhaps how exactly to perform this calculation? If so, I wonder if the question is more appropriate for stats.SE (although I still think it's fine here). $\endgroup$
    – pela
    Commented Jul 6 at 13:56

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The value of $H_0$ as measured by the Planck Satellite is 67.8 km/s/Mpc (https://sci.esa.int/web/planck/-/60504-measurements-of-the-hubble-constant). The standard deviation of the joint analysis is $\sigma \approx 1.75$ km/s/Mpc. So the difference between the mean value of $H_0$ in the joint analysis and the value from Planck is $3.1\sigma$.

As you mentioned, the probability that the value you measured lies within $3 \sigma$ is 99.7%, so the probability that the offset between the different $H_0$ measurements is due to a statistical error is $<0.3\%$.

In general, $k\sigma$ measures the degree of certainty you have in your measurement. If your instrument has a noise level of $\sigma$ and you measure something above $3\sigma$, you can be 99.7% sure that this was actually a signal.

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  • $\begingroup$ Can we determine the value of $k$ from Fisher information matrix? $\endgroup$
    – Wang Yun
    Commented Jul 5 at 11:43

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