In this talk, we will study moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. In conclusion, we will obtain the Sato-Tate distribution for the trace of certain families of Elliptic curves in arithmetic progressions. As a special case we will recover a result of Birch proving Sato-Tate distribution for certain families of elliptic curves. Moreover, we will see that these results follow from asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. Finally, if time permits, we will discuss the bias conjecture in the finite field setting. These are joint works with Kathrin Bringmann, Ben Kane, and Zichen Yang.