Variability, both in X-ray and optical/UV, affects the well-known anti-correlation between the $alpha_{ox}$ spectral index and the UV luminosity of active galactic nuclei, contributing part of the dispersion around the average correlation (intra-source dispersion), in addition to the differences among the time-average $alpha_{ox}$ values from source to source (inter-source dispersion). We want to evaluate the intrinsic $alpha_{ox}$ variations in individual objects, and their effect on the dispersion of the $alpha_{ox}-L_{UV}$ anti-correlation. We use simultaneous UV/X-ray data from Swift observations of a low-redshift sample, to derive the epoch-dependent $alpha_{ox}(t)$ indices. We correct for the host galaxy contribution by a spectral fit of the optical/UV data. We compute ensemble structure functions to analyse variability of multi-epoch data. We find a strong intrinsic $alpha_{ox}$ variability, which makes an important contribution ($sim40%$ of the total variance) to the dispersion of the $alpha_{ox}-L_{UV}$ anti-correlation (intra-source dispersion). The strong X-ray variability and weaker UV variability of this sample are comparable to other samples of low-z AGNs, and are neither due to the high fraction of strongly variable NLS1s, nor to dilution of the optical variability by the host galaxies. Dilution affects instead the slope of the anti-correlation, which steepens, once corrected, becoming similar to higher luminosity sources. The structure function of $alpha_{ox}$ increases with the time lag up to $sim$1 month. This indicates the important contribution of the intermediate-long timescale variations, possibly generated in the outer parts of the accretion disk.
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