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.

中文翻译：

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对两个宇宙学参数随时间变化的测量的方差分析

该分析的目的是确定给定的误差线是否真正代表了两个宇宙学参数的历史汇编中的数值离散：质量涨落幅度（${\sigma }_8$）和哈勃常数（$H_0$）标准宇宙学模型中的参数。为了进行此分析，对过去测量的已编译列表执行了卡方检验。通过对卡方（${\chi }^2$）的数据值 ${\sigma }_8$ （1993年至2019年之间测得60个数据点 ${\chi }^2$ 在182.4和189.0之间），相关联的概率Q极低， $问=1.6\times {10}^{-15}$ 加权平均值和 $问=8.8\times {10}^{-15}$以获得最佳的数据线性拟合。对于${\chi }^2$ 的值 $H_0$ （1976年至2019年期间测得的163个数据点和 ${\chi }^2$ 在480.1和575.7之间），其中 $问=1.8\times {10}^{-33}$ 用于数据的线性拟合 $问=1.0\times {10}^{-47}$数据的加权平均值。总体结论是，与观察到的参数测量值相关的统计误差线被低估了，或者在至少20％的测量值中未正确考虑系统误差。事实是低估了误差线$H_0$ 如此常见也许可以解释明显的4.4$\sigma$ 差异今天正式称为哈勃紧张局势。