  We prove abstract bounds on the Wasserstein and Kolmogorov distances between
non-randomly centered random sums of real i.i.d. random variables with a finite
third moment and the standard normal distribution. Except for the case of mean
zero summands, these bounds involve a coupling of the summation index with its
size biased distribution as was previously considered in \cite{GolRin96} for
the normal approximation of nonnegative random variables. When being
specialized to concrete distributions of the summation index like the Binomial,
Poisson and Hypergeometric distribution, our bounds turn out to be of the
correct order of magnitude.
