We examine the relationship between the mean and the variability of Arctic sea-ice coverage and volume in a large range of climates from globally ice-covered to globally ice-free conditions. Using a hierarchy of two column models and several comprehensive Earth system models, we consolidate the results of earlier studies and show that mechanisms found in simple models also dominate the interannual variability of Arctic sea ice in complex models. In contrast to predictions based on very idealised dynamical systems, we find a consistent and robust decrease of variance and autocorrelation of sea-ice volume before summer sea ice is lost. We attribute this to the fact that thinner ice can adjust more quickly to perturbations. Thereafter, the autocorrelation increases, mainly because it becomes dominated by the ocean water's large heat capacity when the ice-free season becomes longer. We show that these changes are robust to the nature and origin of climate variability in the models and do not depend on whether Arctic sea-ice loss occurs abruptly or irreversibly. We also show that our climate is changing too rapidly to detect reliable changes in autocorrelation of annual time series. Based on these results, the prospects of detecting statistical early warning signals before an abrupt sea-ice loss at a "tipping point" seem very limited. However, the robust relation between state and variability can be useful to build simple stochastic climate models and to make inferences about past and future sea-ice variability from only short observations or reconstructions.