Let $P$ be the self-similar probability corresponding to an iterated function system of contracting similitudes $S_1,\ldots,S_N$ with probabilities $p_1,\ldots,p_N$ and contraction constants $s_1,\ldots, s_N$. Let $D_r$ be the quantization dimension of $P$ of order $r\in(0,+\infty)$, and let $P_r$ be the self-similar probability corresponding to $S_1,\ldots,S_N$ with probabilities $(p_1s_1^r)^{D_r/(r+D_r)},\ldots,(p_Ns_N^r)^{D_r/(r+D_r)}$. It is shown that the quanitzation coefficient of $P$ of order $r$ exists and that the empirical measures on an arbitrary sequence of asymptotically optimal quantizing sets (of order $r$) weakly converges to $P_r$ provided $\log(p_1s_1^r),\ldots,\log(p_Ns_N^r)$ is not arithmetic and $S_1,\ldots,S_N$ satisfies the open set condition.