Endow the space $\mathcal{P}(\mathbb{R})$ of probability measures on $\mathbb{R}$ with a transportation cost $J(μ, ν)$ generated by a translation-invariant convex cost function. For a probability distribution on $\mathcal{P}(\mathbb{R})$ we formulate a notion of average with respect to this transportation cost, called here the Fréchet barycenter, prove a version of the law of large numbers for Fréchet barycenters, and briefly discuss the structure of $\mathcal{P}(\mathbb{R})$ related to the transportation cost $J$.