  We investigate entropic uncertainty relations for two or more binary
measurements, for example spin-$\frac{1}{2}$ or polarisation measurements. We
argue that the effective anti-commutators of these measurements, i.e. the
anti-commutators evaluated on the state prior to measuring, are an expedient
measure of measurement incompatibility. Based on the knowledge of pairwise
effective anti-commutators we derive a class of entropic uncertainty relations
in terms of conditional R\'{e}nyi entropies. Our uncertainty relations are
formulated in terms of effective measures of incompatibility, which can be
certified device-independently. Consequently, we discuss potential applications
of our findings to device-independent quantum cryptography. Moreover, to
investigate the tightness of our analysis we consider the simplest (and very
well-studied) scenario of two measurements on a qubit. We find that our results
outperform the celebrated bound due to Maassen and Uffink [Phys. Rev. Lett. 60,
1103 (1988)] and provide a new analytical expression for the minimum
uncertainty which also outperforms some recent bounds based on majorisation.
