and the relation between $\beta$ and UV absolute magnitude ($\beta$–$M_{\mathrm{UV}}$ relation) has been a subject of debate for the several years (e.g., [@Bouw12]; [@Bouw14]; [@Dunl12]; [@Dunl13]; [@Fink12]; [@Rog14]). and report that bright galaxies have redder $\beta$ values and faint galaxies have bluer $\beta$ values, while and report that $\beta$ values are constant over the observed magnitude range. This inconsistency in the $\beta$–$M_{\mathrm{UV}}$ relation can be caused by both large photometric errors for faint galaxies and selection bias. In order to reveal the true $\beta$–$M_{\mathrm{UV}}$ relation, a further large sample of objects with small photometric uncertainties is needed and/or it is necessary to assess the incompleteness of the observed $\beta$ distribution. Recently, @Dunc15 and @Bouw14 show that the $\beta$ value decreases with the $M_{\mathrm{UV}}$ value. finds the trend by combining the results of the previous literature ([@Bouw14];[@Dunc14]; [@Dunl12]; [@Dunl13]; [@Fink12]; [@Rog14]; [@Wil11]) and discusses the trend by assessing the observational bias (incompleteness) of the observed $\beta$ distribution in the faint magnitude range.

The $\beta$–$M_{\mathrm{UV}}$ relation is understood as another aspect of the mass-metallicity relation seen in star-forming galaxies as the $\beta$ value depends on the dust extinction ([@Bouw09], , ). Moreover, it is suggested that there is a “knee” in the 