  We present a study of the hypothesis that white dwarfs undergo a spectral
change from hydrogen- to helium-dominated atmospheres using a volume-limited
photometric sample drawn from the $Gaia$ DR2 catalogue, the Sloan Digital Sky
Survey (SDSS) and the $Galaxy~Evolution~Explorer$ (GALEX). We exploit the
strength of the Balmer jump in hydrogen-atmosphere DA white dwarfs to separate
them from helium-dominated objects in SDSS colour space. Across the effective
temperature range from 20000K to 9000K we find that 22% of white dwarfs will
undergo a spectral change, with no spectral evolution being ruled out at
5$\sigma$. The most likely explanation is that the increase in He-rich objects
is caused by the convective mixing of DA stars with thin hydrogen layers, in
which helium is dredged up from deeper layers by a surface hydrogen convection
zone. The rate of change in the fraction of He-rich objects as a function of
temperature, coupled with a recent grid of 3D radiation-hydrodynamic
simulations of convective DA white dwarfs - which include the full overshoot
region - lead to a discussion on the distribution of total hydrogen mass in
white dwarfs. We find that 60% of white dwarfs must have a hydrogen mass larger
than $M_{\rm H}/M_{\rm WD} = 10^{-10}$, another 25% have masses in the range
$M_{\rm H}/M_{\rm WD} = 10^{-14}-10^{-10}$, and 15% have less hydrogen than
$M_{\rm H}/M_{\rm WD} = 10^{-14}$. These results have implications for white
dwarf asteroseismology, stellar evolution through the asymptotic giant branch
(AGB) and accretion of planetesimals onto white dwarfs.
