In the present study we investigate the statistical thermodynamics of the anharmonic oscillator, whose energies are characterized by the potential 1/2x 2+x 4. Employing the energies recently obtained by Hioe and Montroll, we compute the partition function and the thermodynamic quantities for the anharmonic and quartic oscillators. Low- and high-temperature formulas are presented for the thermodynamic quantities of the oscillators.