Distributed calibration based on consensus optimization is a computationally efficient method to calibrate large radio interferometers such as LOFAR and SKA. Calibrating along multiple directions in the sky and removing the bright foreground signal is a crucial step in many science cases in radio interferometry. The residual data contain weak signals of huge scientific interest and of particular concern is the effect of incomplete sky models used in calibration on the residual. In order to study this, we consider the mapping between the input uncalibrated data and the output residual data. We derive an analytical relationship between the input and output probability density functions which can be used to study the performance of calibration.