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The people say that the value measured for the Hubble constant for the CMB is 4.4 $\sigma$ away from the value measured by Type Ia supernovae. I don't get it

The value for CMB is $H_{0} = 67.4 \pm 0.5 km s^{−1} Mpc^{−1}$ while for Supernovae $H_{0} = 74.03 \pm 1.42 km s^{−1} Mpc^{−1}$.

If I do some easy calculation (taking the mean value of $H_{0}$ as the real value for $H_{0}$ and $\sigma$ as the value measured by Supernovae )

I get

$$\sigma=\frac{74.03-67.4}{1.42}=4.66$$

I think that is to far from $4.4\sigma$. What I am doing wrong?

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  • $\begingroup$ This is the place where I read about $4.4\sigma$ tension $\endgroup$
    – Nothing
    Commented Feb 9, 2020 at 20:31

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The difference between the two values should be zero if the cosmological model is correct.

The measured difference is $6.63 \pm 1.51$ km/s per Mpc. This assumes the uncertainties in the two measurements can simply be added in quadrature (i.e. $1.51 =\sqrt{1.42^2 + 0.5^2}$). This is slightly different from your result because you didn't include the smaller uncertainty in the first measurement for some reason.

This result differs from zero by $6.63/1.51 = 4.4$ times its uncertainty. Hence we say is is a $4.4\sigma$ result.

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  • $\begingroup$ Regarding: "The difference between the two values should be zero if the cosmological model is correct," there are other possibilities to explain this discrepancy. (1) There is something wrong with the astronomical "yard stick", that may yet turn out to be identified and corrected. (2) There is something wrong with the theory regarding the CMB variability that may yet turn out to be identified and corrected. (3) Various combinations of both (1) and (2) might be found and corrected. $\endgroup$
    – Buzz
    Commented Nov 13, 2020 at 16:59

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