Temperature dependence of magnetic moment (m-T) and the magnetization (M-H) at 5 K and 20 K of the in situ processed $MgB_2$ bulk pellets with/without carbon (C) doping were examined. The superconducting critical temperature ($T_c$), the superconducting transition width (${delta}T$) and the critical current density ($J_c$) were estimated for ten test samples taken from the $MgB_2$ bulk pellets. The reliable m-T characteristics associated with the uniform $MgB_2$ formation were obtained for both $MgB_2$ pellets. The $T_cs$ and ${delta}Ts$ of all test samples of the undoped $MgB_2$ were the same each other as 37.5 K and 1.5 K, respectively. The $T_cs$ and ${delta}Ts$ of the C-doped $MgB_2$ were 36.5 K and 2.5 K, respectively. Unlike the m-T characteristics, there existed the difference among the M-H curves of the test samples, which might be caused by the microstructure variation. In spite of the slight $T_c$ decrease, the C doping was effective in enhancing the $J_c$ at 5 K.