summarized.

\[tab3\]

The standard deviation is clearly larger than the typical uncertainty in the $\beta$ value except for the two of the most faint magnitude bin. Therefore, in the magnitude range of $M_{\mathrm{UV}} \lesssim -20.5$, the observed $\beta$ distribution is more scattered than the typical uncertainty and the observed scatter represents the variation of the stellar population and dust extinction among the sample. In the magnitude range of $M_{\mathrm{UV}} \gtrsim -20.5$, the typical uncertainty in $\beta$ becomes as large as the standard deviation, and particularly the objects with $M_{\mathrm{UV}} \gtrsim -20.0$ are not uniformly distributed. Although we does not add any criteria to our sample since our purpose is to investigate the bright LBGs with *i’* $\leq$ 26.0, we check the sample completeness in section \[S4s2RF\] and discuss the results with some $M_{\mathrm{UV}}$ sub-samples.

The mean $\beta$ value seems not to decrease with the mean UV magnitude. In order to quantify the $\beta$–$M_{\mathrm{UV}}$ trend, we conduct the least-square linear fitting for our mean $\beta$ and $M_{\mathrm{UV}}$ values described in table \[tab3\]. We use the four data points in $-22.0 \leq M_{\mathrm{UV}} \leq -20.0$, and the slope of the fitted linear equation becomes $-0.02\,\pm\,0.02$, which is nearly zero compared with the 