  We study the energy-critical nonlinear Schr\"{o}dinger equation with
randomised initial data in dimensions $d>6$. We prove that the Cauchy problem
is almost surely globally well-posed with scattering for randomised
super-critical initial data in $H^s(\mathbb{R}^d)$ whenever
$s>\max\{\frac{4d-1}{3(2d-1)},\frac{d^2+6d-4}{(2d-1)(d+2)}\}$. The
randomisation is based on a decomposition of the data in physical space,
frequency space and the angular variable. This extends previously known results
of Spitz in dimension 4. The main difficulty in the generalisation to high
dimensions is the non-smoothness of the nonlinearity.
