We carried out NPT-ensemble (constant-number of particles, pressure, and temperature) Molecular Dynamics (MD) simulations for measuring strength anisotropy under uniaxial compressive stress rotated to the crystallographic axes in $alpha$-quartz. Uniaxial compressive strengths of a single quartz crystal were measured in directions of the a- and c-axis. Measured uniaxial strength of a single quartz crystal was higher in the direction parallel to the c-axis than that measured in the direction normal to the c-axis. However the reverse was found in calculated uniaxial strengths by MD simulation. The contradictive result of strengths was observed in both cases but was found to be different in origin. Strength anisotropy of defectless $alpha$-quartz crystal in MD simulation is basically caused by structural difference of quartz. By contrast, anisotropy of measured strength in the uniaxial compression test is related to oriented micro-defects developed during crystal growth.