  Andromeda XXI (And XXI) has been proposed as a dwarf spheroidal galaxy with a
central dark matter density that is lower than expected in the Standard
$\Lambda$ Cold Dark Matter ($\Lambda$CDM) cosmology. In this work, we present
dynamical observations for 77 member stars in this system, more than doubling
previous studies to determine whether this galaxy is truly a low density
outlier. We measure a systemic velocity of $v_r=-363.4\pm1.0\,{\rm kms}^{-1}$
and a velocity dispersion of $\sigma_v=6.1^{+1.0}_{-0.9}\,{\rm kms}^{-1}$,
consistent with previous work and within $1\sigma$ of predictions made within
the modified Newtonian dynamics framework. We also measure the metallicity of
our member stars from their spectra, finding a mean value of ${\rm
[Fe/H]}=-1.7\pm0.1$~dex. We model the dark matter density profile of And~XXI
using an improved version of \GravSphere, finding a central density of
$\rho_{\rm DM}({\rm 150 pc})=2.7_{-1.7}^{+2.7} \times 10^7 \,{\rm
M_\odot\,kpc^{-3}}$ at 68\% confidence, and a density at two half light radii
of $\rho_{\rm DM}({\rm 1.75 kpc})=0.9_{-0.2}^{+0.3} \times 10^5 \,{\rm
M_\odot\,kpc^{-3}}$ at 68\% confidence. These are both a factor ${\sim}3-5$
lower than the densities expected from abundance matching in $\Lambda$CDM. We
show that this cannot be explained by `dark matter heating' since And~XXI had
too little star formation to significantly lower its inner dark matter density,
while dark matter heating only acts on the profile inside the half light
radius. However, And~XXI's low density can be accommodated within $\Lambda$CDM
if it experienced extreme tidal stripping (losing $>95\%$ of its mass), or if
it inhabits a low concentration halo on a plunging orbit that experienced
repeated tidal shocks.
