For a sequence of continuous, monotone functions $$f_1,\ldots ,f_n :I \rightarrow \mathbb {R}$$ f1,…,fn:I→R (I is an interval) we define the mapping $$M :I^n \rightarrow I^n$$ M:In→In as a Cartesian product of quasi-arithmetic means generated by $$f_j$$ fj -s. It is known that, for every initial vector, the iteration sequence of this mapping tends to the diagonal of $$I^n$$ In . We will prove that whenever all $$f_j$$ fj -s are $$\mathcal {C}^2$$ C2 with nowhere vanishing first derivative, then this convergence is quadratic. Furthermore, the limit $$\frac{\text {Var}\, M^{k+1}(v)}{(\text {Var}\, M^{k}(v))^2}$$ VarMk+1(v)(VarMk(v))2 will be calculated in a nondegenerated case.