and the middle-right panel shows the median values. The median values are consistent with zero at the level of $\sim 0.1$ mas/yr, and there is no systematic dependence on $\Delta T$. As expected, there is a strong correlation between the dispersion of QSO proper motions and $\Delta T$. The dashed red line shows a model fit to the relation of the form:

$$\label{eqn:sig}
\sigma = A + B/\Delta T,$$

where $A=0.157$ mas/yr and $B=22.730$ mas. It is encouraging that this simple $A+B/\Delta T$ model agrees well with the QSO data, and we find no significant systematic differences between different SDSS data releases. Note that we show in Appendix \[sec:appendix\] that there is no significant systematic variation in the QSO proper motions with position on the sky.

![image](sdss_rrl.png){width="16cm" height="5.33cm"}

We also use the QSO sample to check whether or not the proper motion uncertainties vary significantly with magnitude or colour. In Fig. \[fig:qso\_mag\_col\] we show the dispersion in QSO proper motions as a function of $r$-band magnitude (left panel) and $g-r$ colour (right panel). The dotted lines indicate the median standard deviation in proper motion of 2 mas/yr. There is a weak dependence on r-band magnitude, whereby the QSO proper motion 