This paper investigates the possibility of improving radio interferometric images using an algorithm inspired by an optical method known as "lucky imaging", which would give more weight to the best-calibrated visibilities used to make a given image. A fundamental relationship between the statistics of interferometric calibration solution residuals and those of the image-plane pixels is derived in this paper. This relationship allows us to understand and describe the statistical properties of the residual image. In this framework, the noise map can be described as the Fourier transform of the covariance between residual visibilities in a new differential Fourier plane. Image-plane artefacts can be seen as one realisation of the pixel covariance distribution, which can be estimated from the antenna gain statistics. Based on this relationship, we propose a means of improving images made with calibrated visibilities using weighting schemes. This improvement would occur after calibration, but before imaging; it is thus ideally used between major iterations of self-calibration loops. Applying the weighting scheme to simulated data improves the noise level in the final image at negligible computational cost.