  The ejected mass distribution of type Ia supernovae directly probes
progenitor evolutionary history and explosion mechanisms, with implications for
their use as cosmological probes. Although the Chandrasekhar mass is a natural
mass scale for the explosion of white dwarfs as type Ia supernovae, models
allowing type Ia supernovae to explode at other masses have attracted much
recent attention. Using an empirical relation between the ejected mass and the
light curve width, we derive ejected masses $M_\mathrm{ej}$ and $^{56}$Ni
masses $M_\mathrm{Ni}$ for a sample of 337 type Ia supernovae with redshifts $z
< 0.7$ used in recent cosmological analyses. We use hierarchical Bayesian
inference to reconstruct the joint $M_\mathrm{ej}$-$M_\mathrm{Ni}$
distribution, accounting for measurement errors. The inferred marginal
distribution of $M_\mathrm{ej}$ has a long tail towards sub-Chandrasekhar
masses, but cuts off sharply above 1.4 $M_\odot$. Our results imply that
25\%-50\% of normal type Ia supernovae are inconsistent with Chandrasekhar-mass
explosions, with almost all of these being sub-Chandrasekhar-mass;
super-Chandrasekhar-mass explosions make up no more than 1\% of all
spectroscopically normal type Ia supernovae. We interpret the type Ia supernova
width-luminosity relation as an underlying relation between $M_\mathrm{ej}$ and
$M_\mathrm{Ni}$, and show that the inferred relation is not naturally explained
by the predictions of any single known explosion mechanism.
