In this talk, I will present a central limit theorem describing the fluctuations of the number of zeros (local statistics) of various families of $L$-functions around their mean. The correlations of these fluctuations coincide with those obtained by Wieand, Diaconis, and Evans for the number of eigenvalues of random matrices. The families of $L$-functions considered here are defined over function fields associated with hyperelliptic curves of large genus over a fixed finite field.