  We revisit the electroweak phase transition in the standard model with a real
scalar, utilizing several calculation methods to investigate scheme
dependences. We quantify the numerical impacts of Nambu-Goldstone resummation,
required in one of the schemes, on the strength of the first-order electroweak
phase transition. We also employ a gauge-independent scheme to make a
comparison with the standard gauge-dependent results. It is found that the
effect of the Nambu-Goldstone resummation is typically $\sim1\%$. Our analysis
shows that both gauge-dependent and -independent methods give qualitatively the
same result within theoretical uncertainties. In either methods, the scale
uncertainties in the ratio of critical temperature and the corresponding Higgs
vacuum expectation value are more than 10%, which signifies the importance of
higher-order corrections.
