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In this article, we study the approximation of a probability measure $\mu$ on ${ℝ}^d$ by its empirical measure ${\widehat{\mu }}_N$ interpreted as a random quantization. As error criterion we consider an averaged $p$th moment Wasserstein metric. In the case where $2plt;d$, we establish fine upper and lower bounds for the error, a. Moreover, we provide a universal estimate based on moments, a . In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.