In this paper I calculate nine statistics on cyclic permutations in one-line notation that avoid two patterns in their cycle notation (namely 231 and 312), including peaks, inversions, descents and idescents, excedances, and left to right and right to left minima and maxima. Furthermore, I prove formulas for the number of such statistics for each permutation of any given n. Finally, I end the paper by proposing similar formulas for the 5 other interesting pairs of patterns, with their corresponding statistics coming in a subsequent paper.