  We analyse the first measurements from precision weak lensing (PWL): a new
methodology for measuring individual galaxy-galaxy weak lensing through
velocity information. Our goal is to understand the observed shear distribution
from PWL, which is broader than can be explained by the statistical measurement
errors. We identify two possible sources of scatter to explain the observed
distribution: a shape noise term associated with the underlying assumption of
circular stable rotation, and an astrophysical signal consistent with a
log-normal dispersion around the stellar-to-halo mass relation (SHMR). We have
modelled the observed distribution as the combination of these two factors and
quantified their most likely values given our data. For the current sample, we
measure an effective shape noise of $\sigma_\gamma = 0.024 \pm 0.007$,
highlighting the low noise impact of the method and positioning PWL as $\sim
10$ times more precise than conventional weak lensing. We also measure an
average dispersion in shears of $\xi_\gamma = 0.53^{+0.26}_{-0.28}$\,dex over
the range of $8.5 < \log M_\star < 11$. This measurement is higher than
expected, which is suggestive of a relatively high dispersion in halo mass
and/or profile.
