A Chi-Squared Analysis of the Measurements of Two Cosmological
  Parameters Over Time
  The aim of this analysis was to determine whether or not the given error bars
truly represented the dispersion of values in a historical compilation of two
cosmological parameters: the amplitude of mass fluctuations ($\sigma_8$) and
Hubble's constant ($H_0$) parameters in the standard cosmological model. For
this analysis, a chi-squared test was executed on a compiled list of past
measurements. It was found through analysis of the chi-squared ($\chi^2$)
values of the data that for $\sigma_8$ (60 data points measured between 1993
and 2019 and $\chi^2$ between 182.4 and 189.0) the associated probability Q is
extremely low, with $Q = 1.6 \times 10^{-15}$ for the weighted average and $Q =
8.8 \times 10^{-15}$ for the best linear fit of the data. This was also the
case for the $\chi^2$ values of $H_0$ (163 data points measured between 1976
and 2019 and $\chi^2$ between 480.1 and 575.7), where $Q = 1.8 \times 10^{-33}$
for the linear fit of the data and $Q = 1.0 \times 10^{-47}$ for the weighted
average of the data. The general conclusion was 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 20\% of
the measurements. The~fact that the underestimation of error bars for $H_0$ is
so common might explain the apparent 4.4$\sigma$ discrepancy formally known
today as the Hubble tension.
