In this paper, we prove the Hyers--Ulam stability of the following generalized additive functional equation \[ \sum_{1\leq i\leq j\leq m}f\bigg(\frac{x_i+x_j}2+\sum_{l=1,k_l\neq i,j}^{m-2}x_{k_l}\bigg)=\frac{(m-1)^2}2\sum_{i=1}^mf(x_i) \] where $m$ is a positive integer greater than 3, in various normed spaces.