  We study the random data problem for 3D, defocusing, cubic nonlinear
Schr\"odinger equation in $H_x^s(\mathbb{R}^3)$ with $s<\frac 12$. First, we
prove that the almost sure local well-posedness holds when
$\frac{1}{6}\leqslant s<\frac 12$ in the sense that the Duhamel term belongs to
$H_x^{1/2}(\mathbb{R}^3)$.
  Furthermore, we prove that the global well-posedness and scattering hold for
randomized, radial, large data $f\in H_x^{s}(\mathbb{R}^3)$ when
$\frac{17}{40}< s<\frac 12$. The key ingredient is to control the energy
increment including the terms where the first order derivative acts on the
linear flow, and our argument can lower down the order of derivative more than
$\frac12$. To our best knowledge, this is the first almost sure large data
global result for this model.
