Summary: Recently, Z.-G. Xiao et al. (see [Math. Comput. Modelling 51, No. 5–6, 592–600 (2010; Zbl 1190.26031)]) have introduced a new refinement of the discrete Jensen’s inequality for mid-convex functions. We give and discuss the weighted form of their results. This leads to some new inequalities and limit formulas. We illustrate the scope of the results by applying them to introduce and study some new quasi-arithmetic means.