  Recently Ono, Saad and the second author \cite{KHN} initiated a study of
value distribution of certain families of Gaussian hypergeometric functions
over large finite fields. They investigated two families of Gaussian
hypergeometric functions and showed that they satisfy semicircular and Batman
distributions. Motivated by their results we aim to study distributions of
certain families of hypergeometric functions in the $p$-adic setting over large
finite fields. In particular, we consider two and six parameters families of
hypergeometric functions in the $p$-adic setting and obtain that their limiting
distributions are semicircular over large finite fields. In the process of
doing this we also express the traces of $p$th Hecke operators acting on the
spaces of cusp forms of even weight $k\geq4$ and levels 4 and 8 in terms of
$p$-adic hypergeometric function which is of independent interest. These
results can be viewed as $p$-adic analogous of some trace formulas of \cite{ah,
ah-ono, fop}.
