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 $-2.21\pm 0.13$, 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, $\alpha OX,Var$; the average for this sample is $\alpha OX,Var=-1.06\pm 0.04$. The anticorrelation of $\alpha OX$ with the UV luminosity, $LUV$, previously found in the average emission is also present in the variable part: $\alpha OX,Var=(-0.177\pm 0.083)log(L\nu ,Var(2500\backslash AA))+(3.88\pm 2.33)$. 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.