We propose a class of incompatibility measures for quantum observables based on quantifying the effect of a measurement of one observable on the statistics of the outcomes of another. Specifically, for a pair of observables A and B with purely discrete spectra, we compare the following two probability distributions: one resulting from a measurement of A followed by a measurement of B on a given state and the other obtained from a measurement of B alone on the same state. We show that maximizing the distance between these two distributions over all states yields a valid measure of the incompatibility of observables A and B, which is zero if and only if they commute and is strictly greater than zero (and less than or equal to one) otherwise. For finite-dimensional systems, we obtain a tight upper bound on the incompatibility of any pair of observables and show that the bound is attained when the observables are totally nondegenerate and associated with mutually unbiased bases. In the process, we also establish an important relation between the incompatibility of a pair of observables and the maximal disturbances due to their measurements. Finally, we indicate how these measures of incompatibility and disturbance can be extended to the more general class of nonprojective measurements. In particular, we obtain a nontrivial upper bound on the incompatibility of one Lüders instrument with another.