The number of points on a certain one parameter family of algebraic surface over a finite field $\F_p$ can be expressed as $p^2+A_p(λ),$ where $A_p(λ)$ is a character sum and $λ$ is an element of the finite field $\F_p.$ In this paper, we study the distribution of the term $A_p(λ)$ as the surface varies over a large family of algebraic surfaces of fixed genus and growing $p.$ The power moments of $A_p$'s are weighted sums of Catalan numbers. As a consequence of these results, we obtain limiting distributions of certain families of hypergeometric functions over large finite fields.