Hubble tensions: a historical statistical analysis

Abstract: Statistical analyses of the measurements of the Hubble-Lema^itre constant $H_0$ (163 measurements between 1976 and 2019) show that the statistical error bars associated with the observed parameter measurements have been underestimated -- or the systematic errors were not properly taken into account -- in at least 15-20\% of the measurements. The fact that the underestimation of error bars for $H_0$ is so common might explain the apparent discrepancy of values, which is formally known today as the Hubble tension. Here we have carried out a recalibration of the probabilities with this sample of measurements. We find that $x\sigma $ deviation is indeed equivalent in a normal distribution to $x_{\rm eq.}\sigma $s deviation in the frequency of values, where $x_{\rm eq.}=0.83x^{0.62}$. Hence, a tension of 4.4$\sigma $, estimated between the local Cepheid-supernova distance ladder and cosmic microwave background (CMB) data, is indeed a 2.1$\sigma $ tension in equivalent terms of a normal distribution of frequencies, with an associated probability $P(>x_{\rm eq.})=0.036$ (1 in 28). This can be increased up to a equivalent tension of 2.5$\sigma $ in the worst of the cases of claimed 6$\sigma $ tension, which may anyway happen as a random statistical fluctuation.