  With the advent of Integral Field Units (IFUs), surveys can now measure
metallicities across the discs of nearby galaxies at scales $\lesssim 100~$ pc.
At such small scales, many of these regions contain too few stars to fully
sample all possible stellar masses and evolutionary states, leading to
stochastic fluctuations in the ionising continuum. The impact of these
fluctuations on the line diagnostics used to infer galaxy metallicities is
poorly understood. In this paper, we quantify this impact for six most
commonly-used diagnostics. We generate stochastic stellar populations for
galaxy patches with star formation rates varying over a factor of $1000$,
compute the nebular emission that results when these stars ionise gas at a wide
range of densities, metallicities, and determine how much inferred
metallicities vary with fluctuations in the driving stellar spectrum. We find
that metallicities derived from diagnostics that measure multiple ionisation
states of their target elements (e.g.~electron temperature methods) are weakly
affected (variation $< 0.1$ dex), but that larger fluctuations ($\sim 0.4$ dex)
occur for diagnostics that depend on a single ionisation state. Scatter in the
inferred metallicity is generally largest at low star formation rate and
metallicity, and is larger for more sensitive observations than for shallower
ones. The main cause of the fluctuations is stochastic variation in the
ionisation state in the nebula in response to the absence of Wolf-Rayet stars,
which dominate the production of $\gtrsim 2-3$ Ryd photons. Our results
quantify the trade-off between line brightness and diagnostic accuracy, and can
be used to optimise observing strategies for future IFU campaigns.

Astrophysics of Galaxies