This is joint work with N. Prabhu (Queens University at Kingston, Canada). We derive new bounds for moments of the error in the Sato-Tate law over families of elliptic curves. As applications, we deduce new almost-all results for the said errors and a conditional Central Limit Theorem on the distribution of these errors. Our method builds on recent work by N. Prabhu and K. Sinha who derived a Central Limit Theorem on the distribution of the errors in the Sato-Tate law for families of cusp forms for the full modular group. In addition, identities by Birch and Melzak play a crucial rule in this paper. Birch's identities connect moments of coefficients of Hasse-Weil $L$-functions for elliptic curves with the Kronecker class number and further with traces of Hecke operators. Melzak's identity is combinatorial in nature. If time permits, we shall also talk about ongoing work with N. Prabhu and K. Sinha in which progress is made via a smoothing.