Using first-principles calculations, we examine the transition temperature T c of superconductivity in sodium tungsten bronze ( Na x WO 3 , where x is equal to or less than unity ). Although T c is relatively low T c ( < ∼ 3K), it is interesting that its characteristic exponential dependence on x has been experimentally observed at 0.2 < ∼ x < ∼ 0.4. On the basis of the McMillan equation for T c including the effect of plasmons, we succeed in reproducing the absolute values of T c and its x dependence. We also find that the plasmon effect is crucial for the estimation of T c as well as phonons. Since the calculated T c may not exceed ∼ 20 K even for x < ∼ 0.1, the superconductivity at a low T c can be interpreted by the usual phonon mechanism, including the plasmon effect. On the other hand, a high T c up to about 90 K, which is found on the surface of a Na x WO 3 system at x ∼ 0.05 by recent experiments, cannot be explained by our results. This discrepancy suggests that another mechanism is required to clarify the nature of the high-T c superconductivity of Na x WO 3 .