Abstract Let {Xn;n1} be a sequence of i.i.d. random variables and let X(r)n = Xj if |Xj| is the r-th maximum of |X1|, ..., |Xn|. Let Sn = X1++Xn and (r)Sn = Sn(X(1)n++X(r)n). Sufficient and necessary conditions for (r)Sn approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for (r)Sn are studied.