i)-\bar{y})^2}$$ where $y_{i}$ is the value of $\mu_\alpha\cos\delta (\mu_\delta$) for each star, $i$, $\theta$ is the corresponding position angle, $\bar{y}$ is the average value of $y$, and $f$ is the best-fit function. The $r^{2}$ values for Fe-poor and Fe-rich are listed in bottom left insets of Figure \[fig:rot\]. The fact that the motions of $\omega\,$Centauri and M22 provide a good match between the data and the sine interpolation ($r^{2}>0.7$), confirms the visual impression that both populations rotate in the plane of the sky.

Figure \[fig:rot\] shows that the sine functions that provide the best fit of Fe-poor and Fe-rich stars exhibit slight different rotation patterns. To investigate whether these differences are significant or not, we followed the procedure introduced by @cordoni2019 [see their Section 4.1]. Specifically, we ran 1,000 Monte Carlo simulations of Fe-poor and Fe-rich stars where we assumed that the two populations exhibit the same proper motion distribution corresponding to the sine function that best fits the observed Fe-poor stars. We assumed that the two populations host the same number of stars as the observed Fe-rich and Fe-poor stars and added the corresponding observational errors to the simulated proper motions of each star. We used Equation 2 to 