Tension parallel to the radial direction (RDT) and compression along the extrusion direction (EDC) of an extruded AZ31 Mg alloy bar were conducted under a strain rate of 10^-4 s^-1 at room temperature. Twinning behavior was characterized by optical microscopy (OM) and electron backscatter diffraction (EBSD). The results demonstrate that extension twins have an effect on the yield point; the yield point of the alloy under EDC is 139 MPa which is higher than 88 MPa of the alloy under RDT. With increasing strain, strain hardening rates under these two stress states show a trend of fast decline at first, and then the hardening rates under EDC start to increase obviously till fracture while that under RDT keeps almost a constant. The increase in the hardening rate under EDC is related to the formation of extension twins and the twinning texture-induced slip activity. A method to measure the twin volume fraction in grains based on EBSD results was proposed. Under RDT, the volume fraction of extension twin in (0002) grains is about 45% at the strain of 0.04.