  We derive new bounds for moments of the error in the Sato-Tate law over
families of elliptic curves. Our estimates are stronger than those obtained by
W.D. Banks and I.E. Shparlinski (arXiv:math/0609144) and L. Zhao and the
fist-named author in (arXiv:math/0608318) for the first and second moments, but
this comes at the cost of larger ranges of averaging. 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 is different from
those used in the above-mentioned papers and builds on recent work by the
second-named author and K. Sinha (arXiv:1705.04115) 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.
