The distance distribution of an extended (binary) Goppa code of length \(N\) with Goppa polynomial \(G(x)\in \mathbb{F}_2[x]\) of degree \(t\) is approximately binomial, i.e., the number \(B_{2k}\) of codewords of weight \(2k\) is \[ B_{2k}=\frac{{N\choose 2k}}{2^{mt}}(1+E_{2k}), \] with an error term \(E_{2k}\) which tends to zero when \(N\) increases. (Note that \(B_i=0\) for \(i\) odd.) Improving earlier bounds of \textit{S. Vladut} and \textit{A. Skorobogatov} [Probl. Inf. Transm. 27, 19-29 (1991); translation from Probl. Peredachi Inf. 27, 24-36 (1991; Zbl 0732.94012)] and \textit{F. Levy-dit-Vehel} and \textit{S. Litsyn} [Parameters of Goppa codes revisited, IEEE Trans. Inf. Theory 43, 1811-1819 (1997; Zbl 1053.94562)] the authors obtain new bounds on \(E_{2k}\).