We study the energy-critical nonlinear Schrodinger 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 supercritical initial data in H-s(Rd) whenever s > max{4d-1/3(2d-1), d2+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 in dimension 4 [18]. The main difficulty in the generalisation to high dimensions is the non-smoothness of the nonlinearity.