We present an explicit demonstration that a determination of the strong coupling constant $\alpha_s(m_Z)$ from deep-inelastic scattering and hadron collider data without a simultaneous determination of the parton distribution functions (PDFs) leads to a biased result for both the central value and the uncertainty, even in the ideal scenario (closure test) where there are no internal tensions between datasets and where theoretical calculations describe perfectly the experimental measurements. Specifically, we show that a determination of $\alpha_s(m_Z)$ from a single process leads in general to a result that differs from the global best fit more than the value of $\alpha_s(m_Z)$ that is actually favoured by this process.