We consider effects of zero-mean additive noise on systems that are undergoing supercritical blowout bifurcation, i.e., where a chaotic attractor in an invariant subspace loses transverse stability to a nearby on-off intermittent attractor. We concentrate on the low noise limit and two statistical properties of the trajectories; the variance of the normal component and the mean first crossing time of the invariant subspace. Before blowout we find that the asymptotic variance scales algebraically with the noise level and exponentially with the Lyapunov exponent. After blowout it is limited to the nonzero variance of the associated on-off intermittent state. Surprisingly, for a large enough Lyapunov exponent, the effect of added noise can be to decrease rather than increase the variance. The mean crossing time becomes infinite at and after the blowout in the limit of small noise; after the blowout there is exponential dependence on the Lyapunov exponent and algebraic dependence on the noise level. The results are obtained using a drift-diffusion model of Venkataramani et al. The results are confirmed in numerical experiments on a smooth mapping. We observe that although there are qualitative similarities between bubbling (noise-driven) and on-off intermittency (dynamics-driven), these can be differentiated using the statistical properties of the variance of the normal dynamics and the mean crossing time of the invariant subspace in the limit of low noise.