A local and deterministic model of quantum correlations is always possible, as shown explicitly by Brans in 1988: one simply needs the physical systems being measured to have a suitable statistical correlation with the physical systems performing the measurement, via some common cause. Hence, to derive no-go results such as Bell inequalities, an assumption of measurement independence is crucial. It is a surprisingly strong assumption—less than 1 / 15 bits of prior correlation suffice for a local model of the singlet state of two qubits—with ramifications for the security of quantum communication protocols. Indeed, without this assumption, any statistical correlations whatsoever—even those which appear to allow explicit superluminal signalling—have a corresponding local deterministic model. It is argued that ‘quantum nonlocality’ is bad terminology, and that measurement independence does not equate to ‘experimental free will’. Brans’ 1988 model is extended to show that no more than 2logd bits of prior correlation are required for a local deterministic model of the correlations between any two d-dimensional quantum systems.