This work is devoted to the study of rates of convergence of the empirical measures \(\mu_{n} = \frac {1}{n} \sum_{k=1}^n \delta_{X_k}\), \(n \geq 1\), over a sample \((X_{k})_{k \geq 1}\) of independent identically distributed real-valued random variables towards the common distribution \(\mu\) in Kantorovich transport distances \(W_p\). The focus is on finite range bounds on the expected Kantorovich distances \(\mathbb{E}(W_{p}(\mu_{n},\mu ))\) or \(\big [ \mathbb{E}(W_{p}^p(\mu_{n},\mu )) \big ]^1/p\) in terms of moments and analytic conditions on the measure \(\mu \) and its distribution function. The study describes a variety of rates, from the standard one \(\frac {1}{\sqrt n}\) to slower rates, and both lower and upper-bounds on \(\mathbb{E}(W_{p}(\mu_{n},\mu ))\) for fixed \(n\) in various instances. Order statistics, reduction to uniform samples and analysis of beta distributions, inverse distribution functions, log-concavity are main tools in the investigation. Two detailed appendices collect classical and some new facts on inverse distribution functions and beta distributions and their densities necessary to the investigation.