  In this article, we study the approximation of a probability measure $\mu$ on
$\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a
random quantization. As error criterion we consider an averaged $p$-th moment
Wasserstein metric. In the case where $2p<d$, we establish refined upper and
lower bounds for the error, a high-resolution formula. Moreover, we provide a
universal estimate based on moments, a so-called Pierce type estimate. In
particular, we show that quantization by empirical measures is of optimal order
under weak assumptions.
