  We report on a search for low-frequency radio variability in 944 bright (>
4Jy at 154 MHz) unresolved, extragalactic radio sources monitored monthly for
several years with the Murchison Widefield Array. In the majority of sources we
find very low levels of variability with typical modulation indices < 5%. We
detect 15 candidate low frequency variables that show significant long term
variability (>2.8 years) with time-averaged modulation indices M = 3.1 - 7.1%.
With 7/15 of these variable sources having peaked spectral energy
distributions, and only 5.7% of the overall sample having peaked spectra, we
find an increase in the prevalence of variability in this spectral class. We
conclude that the variability seen in this survey is most probably a
consequence of refractive interstellar scintillation and that these objects
must have the majority of their flux density contained within angular diameters
less than 50 milli-arcsec (which we support with multi-wavelength data). At 154
MHz we demonstrate that interstellar scintillation time-scales become long
(~decades) and have low modulation indices, whilst synchrotron driven
variability can only produce dynamic changes on time-scales of hundreds of
years, with flux density changes less than one milli-jansky (without
relativistic boosting). From this work we infer that the low frequency
extra-galactic southern sky, as seen by SKA-Low, will be non-variable on
time-scales shorter than one year.
