We present high sensitivity polarimetric observations in 6 bands covering the 5.5-38 GHz range of a complete sample of 53 compact extragalactic radio sources brighter than 200 mJy at 20 GHz. The observations, carried out with the Australia Telescope Compact Array (ATCA), achieved a 91% detection rate (at 5 sigma). Within this frequency range the spectra of about 95% of sources are well fitted by double power laws, both in total intensity and in polarisation, but the spectral shapes are generally different in the two cases. Most sources were classified as either steep- or peaked-spectrum but less than 50% have the same classification in total and in polarised intensity. No significant trends of the polarisation degree with flux density or with frequency were found. The mean variability index in total intensity of steep-spectrum sources increases with frequency for a 4-5 year lag, while no significant trend shows up for the other sources and for the 8 year lag. In polarisation, the variability index, that could be computed only for the 8 year lag, is substantially higher than in total intensity and has no significant frequency dependence.
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