We analyse a sample of 21 active galactic nuclei (AGN) using data from the Swift satellite to study the variability properties of the population in the X-ray, UV and optical band. We find that the variable part of the UV-optical emission has a spectrum consistent with a powerlaw, with an average index of −2.21±0.13, as would be expected from central illumination of a thin disc (index of −7/3). We also calculate the slope of a powerlaw from UV to X-ray variable emission, α OX,Var ; the average for this sample is α OX,Var = −1.06 ± 0.04. The anticorrelation of α OX with the UV luminosity, L UV , previously found in the average emission is also present in the variable part: α OX,Var = (−0.177 ± 0.083) log(L ν,Var (2500Å))+ (3.88 ± 2.33). Correlated variability between the emission in X-rays and UV is detected significantly for 9 of the 21 sources. All these cases are consistent with the UV lagging the X-rays, as would be seen if the correlated UV variations were produced by the reprocessing of X-ray emission. The observed UV lags are tentatively longer than expected for a standard thin disc.