Threshold magnitude (M₀) is an important factor in determining the magnitude of completeness in calculating seismic b values. Seismic b values can assist in determining the likelihood of earthquake events; therefore, seismic b values should be re-determined repeatedly with ever increasing precision. In this study, we use a median-based analysis of the segment slope (MBASS) to detect change points in Gutenberg-Richter frequency-magnitude distributions (FMDs) to determine M₀. Results give the b value for M₀ - 0.5 to be smaller than the b-value for M₀ + 0.5, and the difference in b values between M₀ and M₀ - 0.5 is larger than the difference in b values between M₀ and M₀ + 0.5. Therefore, b values resulting from M₀ + 0.5 should therefore be more accurate than b values from M₀ - 0.5 when calculating b values using the threshold magnitude (M₀). This is especially true when earthquake events are few and M₀ is large such as for the time periods 1900 - 1935, and 1936 - 1972 as noted in of Taiwan's earthquake catalog.