Bottom-gate tin oxide ($SnO_2$) thin film transistors (TFTs) were fabricated on $N^+$ Si wafers used as gate electrodes. 60-nm-thick $SnO_2$ thin films acting as active layers were sputtered on $SiO_2/Al_2O_3$ films. The $SiO_2/Al_2O_3$ films deposited on the Si wafers were employed for gate dielectrics. In order to increase the resistivity of the $SnO_2$ thin films, oxygen mixed with argon was introduced into the chamber during the sputtering. The mobility of $SnO_2$ TFTs was measured as a function of the flow ratio of oxygen to argon ($O_2/Ar$). The mobility variation with $O_2/Ar$ was analyzed through studies on crystallinity, oxygen binding state, optical properties. X-ray diffraction (XRD) and XPS (X-ray photoelectron spectroscopy) were carried out to observe the crystallinity and oxygen binding state of $SnO_2$ films. The mobility decreased with increasing $O_2/Ar$. It was found that the decrease of the mobility is mainly due to the decrease in the polarizability of $SnO_2$ films.