  We prove a quantitative estimate with a power saving error term for the
number of points in a mapping class group orbit of Teichm\"uller space that lie
within a Teichm\"uller metric ball of given center and large radius. Estimates
of the same kind are also proved for sector and bisector counts. These
estimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin,
and Mirzakhani.
