and tables is that the median value of $A_{200}$ is very robust and only varies between $0.98$ and $1.34$ for our full range of sample selection criteria. With our preferred cuts the median values are $1.15$ and $0.99$ for the narrow and wide $V_{max}$ samples respectively. The best estimate of the true mass of the Local Group (for this definition) is thus very similar to its Timing Argument mass estimate, and depends very little on the calibrating sample of simulated pairs.

The second important point is that the width of the distribution of $A_{200}$ does depend on how the calibrating sample is defined. In particular, it is narrower for samples with the more restrictive $V_{max}$ range, and for given $V_{max}$ range it is smallest for samples with our preferred cuts, those which match the dynamical and morphological properties of the Local Group most closely. For the narrow $V_{max}$ sample the interquartile range is a factor of just $1.6$, and the upper and lower 5% points are separated by a factor of $3$. For the wider velocity range the interquartile range is a factor of $1.8$ and the 5% points are separated by a factor of $5.7$. This shows the Timing 