We study statistical properties of a family of piecewise linear monotone circle maps ft (x) related to the Σ angle doubling map x → 2x mod 1. In particular, we investigate whether for large n, the deviations n−1 i=0 (f it (x0) − 12) upon rescaling satisfies a Q-Gaussian distribution if x0 and t are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if ft is the rotation by t, then it was recently found that in this case the rescaled deviations are distributed as a Q-Gaussian with Q = 2 (a Cauchy distribution). This is the only case where a non-trivial (i.e. Q ̸= 1) Q-Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, limn Sn/n converges to a random variable with a curious distribution which is clearly not a Q-Gaussian or any other standard smooth distribution.