star-forming galaxies (e.g., [@Bouw09], ; [@Mada14]). Currently, the $\beta$–*z* relation has been used for considering the source of cosmic reionization, assuming that the $\beta$ value represents the production rate of hydrogen ionizing photons since the production rate is susceptible to the stellar population (e.g., [@Dunc15]; [@Bouw15], ).

We note that the samples in most of the literature are overlapped and not independent since the set of GOODS-South/HST or HUDF/HST data is mostly used so far in their works. Due to the small observed area, the number of bright objects at $z = 4$ in the field is limited (except for [@Rog14]) and these previous studies focus on relatively faint galaxies at high redshift. The $\beta$ distribution of luminous objects is, however, also important since such population provides important clues to understand early star-formation history in the universe (e.g., [@Cucc12]; [@Hatf18]). In fact, by using stacked images, @LeeKS11 investigate the $\beta$ value of ultra-luminous star-forming galaxies ($19.46 \leq I < 24.96$) at *z* $\sim$ 4 and they find that the $\beta$ value of the stacked star-forming galaxies tends to be redder toward the brighter magnitude range. When taking all the previous works into consideration, the investigation for individual and ”normal” luminous 