We investigate the particle-number fluctuations due to the collective excitations created in a two-dimensional (2D) and a quasi-2D Bose-Einstein condensates (BECs) at low temperature. We find that the fluctuations display an anomalous behavior, i.e., for the 2D BEC they are proportional to
N2, where N is the total number of particles. For the quasi-2D BEC, the particle-number fluctuations are proportional not only to
N2 but also to the square root of the trapping frequency in the strongly confined direction.