​This paper provides a proof of convergence of the standard EnKF generalized to non-Gaussian state space models, based on induction following the proof of Legland 2011 [LGMT+11]. A density-based deterministic approximation of the mean-field EnKF (MFEnKF) is proposed, consisting of a PDE solver and a quadrature rule. Given a certain minimal order of convergence k between the two, this extends to the deterministic filter approximation, which is therefore asymptotically superior to standard EnKF for d<2k. The fidelity of approximation of the true distribution is also established using an extension of total variation metric to random measures. This is limited by a Gaussian bias term arising from non-linearity/non-Gaussianity of the model, which arises in both deterministic and standard EnKF. Numerical results support and extend the theory.