The first data study is presented of possible kinematic biases in the measurement of the top-quark mass based on the invariant mass of its decay products. The analysis uses a sample of ttbar candidate events with one electron or muon and at least four jets in the final state, collected by CMS in pp collisions at sqrt(s)=7 TeV at the LHC. A total of 5174 candidate events is selected from data corresponding to an integrated luminosity of 5.0 fb−1. For each event the mass is reconstructed from a kinematic fit of the decay products to a ttbar hypothesis. The top-quark mass is determined simultaneously with the jet energy scale, constrained by the known mass of the W boson in qqbar decays, in dependence of each kinematic observable. The effects observed are well reproduced by simulations based on MADGRAPH, POWHEG, and MC@NLO with tunes including and excluding the effects of colour reconnection. A measurement is also presented of the top-antitop mass difference. The analysis is based on sqrt(s)=8 TeV data, corresponding to an integrated luminosity of 18.92±0.76 fb-1, and yields a value of Δmt=−272±196 \stat±122 \syst\MeV that is consistent with the standard model prediction. The accuracy of this result is significantly higher than that of existing measurements.