Over sea ice in winter, the clouds, the surface-layer air temperature, and the long-wave radiation are closely coupled. Here we use archived data from the Russian North Pole (NP) drifting stations and our own data from Ice Station Weddell (ISW) to investigate this coupling. Both Arctic and Antarctic distributions of total cloud amount are U-shaped: that is, observed cloud amounts are typically either 0-2 tenths or 8-10 tenths in the polar regions. We fitted these data with beta distributions and, using roughly 70 station-years of observations from the NP stations, compute fitting parameters for each winter month. Although we find that surface-layer air temperature and total cloud amount are correlated, it is not straightforward to predict one from the other because temperature is normally distributed while cloud amount has a U-shaped distribution. Nevertheless, we develop a statistical algorithm that can predict total cloud amount in winter from surface-layer temperature alone and, as required, produces a distribution of cloud amounts that is U-shaped. Because sea ice models usually need cloud data to estimate incoming long-wave radiation, this algorithm may be useful for estimating cloud amounts and, thus, for computing the surface heat budget where no visual cloud observations are available but temperature is measured - from the Arctic buoy network or from automatic weather stations, for example. The incoming long-wave radiation in sea ice models is generally highly parameterized. We evaluate five common parameterizations using data from NP-4, NP-25, and ISW. The formula for estimating incoming long-wave radiation that Konig-Langlo and Augstein developed using both Arctic and Antarctic data has the best properties but does depend nonlinearly on total cloud amount. This nonlinearity is crucial since cloud distributions are U-shaped while common sources of cloud data tabulate only mean monthly values. Lastly, we therefore use a one-dimensional sea ice model to investigate how methods of averaging cloud amounts affect predicted sea ice thickness in the context of the five long-wave radiation parameterizations. Here, too, Konig-Langlo and Augstein's formula performs best, and using daily averaged cloud data yields more realistic results than using monthly averaged cloud data that have been interpolated to daily values.