Most binaries are undetected. Astrometric reductions of a system using the assumption that the object moves like a single point mass can be biased by unresolved binary stars. The discrepancy between the centre of mass of the system (which moves like a point mass) and the centre of light (which is what we observe) introduces additional motion. We explore the extent to which binary systems affect single object model fit to astrometric data. This tells us how observations are diluted by binaries and which systems cause the largest discrepancies – but it also allows us to make inferences about the binarity of populations based on observed astrometric error. By examining a sample of mock observations, we show that binaries with periods close to 1 yr can mimic parallax and thus bias distance measurements, while long-period binaries can introduce significant apparent proper motion. While these changes can soak up some of the error introduced by the binary, the total deviation from the best-fitting model can be translated into a lower limit on the on-sky separation of the pair. Throughout, we link these predictions to data from the Gaia satellite, while leaving the conclusions generalizable to other surveys.