The authors study a Korovkin-type approximation theorem using the equi-statistical convergence which is stronger than the statistical uniform convergence. By an example it is shown that the new approximation result works while its classical and statistical cases do not work. Furthermore, the rate of equi-statistical convergence of a sequence of positive linear operators is computed, and a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials is given.