When one part of the body exerts force on another, the resulting tactile sensation is perceived as weaker than when the same force is applied by an external agent. This phenomenon has been studied using a force matching task, in which observers are first exposed to an external force on a passive finger and then instructed to reproduce the sensation by directly pressing on the passive finger with a finger of the other hand: healthy participants consistently exceed the original force level. However, this exaggeration of the target force is not observed if the observer generates the matching force indirectly, by adjusting a joystick or slider that controls the force output of a motor. Here we present the first formal computational account of this task, in which sensory signals are attenuated based on motor predictions. The modelling elucidates previously unappreciated contributions of multiple sources of noise, including memory noise, in determining matching force output. We show that the predictive component can be isolated by quantifying attenuation as the discrepancy between direct and indirect self-generated forces, rather than direct versus externally generated forces. Our computational account makes the novel prediction that attenuated sensations will display greater trial-to-trial variability than unattenuated ones, because they incorporate additional noise from motor prediction. Quantitative model fitting of force matching data based on close to 500 participants confirmed the prediction of excess variability in self-generated forces and provided evidence for a divisive rather than subtractive mechanism of attenuation, while highlighting its predictive nature.