Noise-induced statistical periodicity in a class of one-dimensional maps is studied. We show theexistence of statistical periodicity in a modified Lasota-Mackey map and describe the phenomenonin terms of almost cyclic sets. A transition from a stable state to a periodic state of the densitydepending on the noise level is observed in numerical investigations based on trajectory averagesand by means of a transfer operator approach. We conclude that the statistical periodicity is theorigin of the almost periodicity in noise-induced order.