  In this note we present new asymptotic estimates comparing the word length
and geodesic length of closed geodesics on surfaces with (variable) negative
sectional curvatures. In particular, we provide an averaged comparison of these
two important quantities and obtain precise statistical results, including a
central limit theorem and a local limit theorem. Further, as a corollary we
also improve an asymptotic formula of R. Sharp and the second author. Finally,
we relate our results to recent work of Gekhtman, Taylor and Tiozzo.
