We investigate the properties of a laser-plasma electron accelerator as a bright source of keV x-ray radiation. During the interaction, the electrons undergo betatron oscillations and from the carefully measured x-ray spectrum the oscillation amplitude of the electrons can be deduced which decreases with increasing electron energies. From the oscillation amplitude and the independently measured x-ray source size of (1.8±0.3) μm we are able to estimate the electron bunch diameter to be (1.6±0.3) μm.