Chi-Squared Analysis of Measurements of Two Cosmological Parameters Over Time

Abstract: For this project, a historical statistical analysis of the Amplitude of Mass Fluctuations ($\sigma_8$) and Hubble's Constant ($H_0$) parameters in the Standard Cosmological Model was carried out to determine whether or not the given error bars truly represent the dispersion of values. It was found through analysis of the Chi-Squared ($\chi^2$) values of the data that for $\sigma_8$ (60 data points and $\chi^2$ between 183.167 and 189.037) that the associated probability Q is extremely low, with $Q = 1.5597'10^{-15}$ for the weighted average and $Q = 1.2107'10^{-14}$ for the best fit of the data. This was also the case for the $\chi^2$ values (163 data points and $\chi^2$ between 484.3977 and 575.655) of $H_0$, where $Q = 4.2176'10^{-34}$ for the linear fit of the data and $Q = 1.0342'10^{-47}$ for the weighted average of the data. Through further analysis, it is shown in question, a linear fit is a better estimate of the data than the weighted average. The general conclusion is that the statistical error bars have been underestimated (in around 20\% of the measurements), or the systematic errors were not properly taken into account.