  We select 947 star-forming galaxies from SDSS-DR7 with
  [O~{\sc iii}]$\lambda$4363 emission lines
  detected at a signal-to-noise {ratio }larger than 5$\sigma$.
  Their electron temperatures and direct oxygen abundances are {then
}determined. {W}e
  compare the results from different methods. $t_2${, the} electron
  temperature in {the }low ionization region{,} estimated from $t_3${, that} in
{the }high ionization region{,} {is} compared {using} three analysis
  relations between $t_2-t_3${. These} show obvious differences, which
  result in some different ionic oxygen abundances. The results of
  $t_3$, $t_2$, {$\rm O^{++}$/$\rm H^+$} and {$\rm O^{+}$/$\rm H^+$} derived by
using
  methods from IRAF and literature are also compared. The ionic
  abundances $\rm O^{++}$/$\rm H^+$ {are} higher than $\rm O^{+}$/$\rm H^+$ for
most
  cases. The{ different} oxygen abundances derived from $T_{\rm e}$ and the
strong-line ratios show {a }clear discrepancy, which is more obvious
  following increasing stellar mass and strong-line ratio
  $R_{23}$. The sample{ of} galaxies from SDSS {with}
  detected [O~{\sc iii}]$\lambda$4363 have lower metallicites and
  higher {star formation rates}, {so} they may not be typical representatives
of the
  whole{ population of} galaxies. Adopting data objects from
  {Andrews \& Martini}, {Liang et al.} and {Lee et al.} data, we derive new
relations of stellar mass and metallicity for star-forming
  galaxies in a much wider stellar mass range: from $10^6\,M_\odot$
  to $10^{11}\,M_\odot$.
