We apply Fourier analysis technique to investigate superconvergence properties of the direct disontinuous Galerkin (DDG) method, DDG method with interface correction (DDGIC), symmetric DDG method and nonsymmetric DDG method. We also include the study of interior penalty DG (IPDG) method, due to its close relation to DDG methods. Error estimates are carried out for both P^2 and P^3 polynomial approximations. By investigating the quantitative errors at the Lobatto points, we show that DDGIC and symmetric DDG methods are superior, in the sense of obtaining (k+2)th superconvergence orders for both P^2 and P^3 approximations. Superconvergence order of (k+2)is also observed for IPDG method with P^3 polynomial approximations. The errors are sensitive to the choice of numerical flux coefficient for even degree P^2 approximations, but are not for odd degree P^3 approximations.Numerical experiments are carried out at the same time and the numerical errors match well with the analytically estimated errors.