from the same distribution. The p-values of all the test are $>> 5\,\%$, and the null hypothesis is not rejected at 5$\%$ significance level. Therefore, there is no evidence that the information about 3.6$\>\micron$ and 4.5$\>\micron$ influences the $\beta$–$M_{\mathrm{UV}}$ relation.

The magenta open circles with the error bar indicate the mean $\beta$ value and the mean $M_{\mathrm{UV}}$ value for each magnitude bin. The standard deviation of the $\beta$ distribution is indicated by the thick marks toward the left side, and the typical uncertainty in the $\beta$ value for the individual objects is shown by the thick marks toward the right side. For the mean values, we just apply the simple geometric mean without taking account of the individual uncertainty in $\beta$ for the individual objects. This is because the mean $\beta$ value can be biased toward positive values if we weight the $\beta$ values by the individual uncertainty of the objects, i.e., the uncertainty is not symmetric and becomes smaller toward positive $\beta$ values than the opposite. The mean $\beta$ value, standard deviation, typical uncertainty, and mean $M_{\mathrm{UV}}$ value for each bin are summarized in table \[tab3\]. In addition, the other useful information such as the median values is also 