We present a study of 9 242 spectroscopically confirmed quasars with multiepoch ugriz photometry from the SDSS Southern Survey. By fitting a separable linear model to each quasar's spectral variations, we decompose their five-band spectral energy distributions into variable (disc) and non-variable (host galaxy) components. In modelling the disc spectra, we include attenuation by dust on the line of sight through the host galaxy to its nucleus. We consider five commonly used attenuation laws, and find that the best description is by dust similar to that of the Small Magellanic Cloud, inferring a lack of carbonaceous grains from the relatively weak 2175-angstrom absorption feature. We go on to construct a composite spectrum for the quasar variations spanning 700-8000 angstrom. By varying the assumed power-law L-nu proportional to nu(alpha) spectral slope, we find a best-fitting value alpha = 0.71 +/- 0.02, excluding at high confidence the canonical L-nu proportional to nu(1/3) prediction for a steady-state accretion disc with a T proportional to r(-3/4) temperature profile. The bluer spectral index of the observed quasar variations instead supports the model of Agol & Krolik, and Mummery & Balbus, in which a steeper temperature profile, T proportional to r(-7/8), develops as a result of finite magnetically induced stress at the innermost stable circular orbit extracting energy and angular momentum from the black hole spin.