  We study the relationship between the gas column density (Sigma_HI) and the
star formation rate surface density (Sigma_SFR) for a sample of extremely small
(M_B ~ -13, Delta V_50 ~ 30 km/s) dwarf irregular galaxies. We find a clear
stochasticity in the relation between the gas column density and star
formation. All gas with Sigma_HI >~ 10 M_sun/pc^2 has some ongoing star
formation, but the fraction of gas with ongoing star formation decreases as the
gas column density decreases, and falls to about 50% at Sigma_HI ~ 3
M_sun/pc^2. Further, even for the most dense gas, the star formation efficiency
is at least a factor of ~ 2 smaller than typical of star forming regions in
spirals. We also find that the ratio of H-alpha emission to FUV emission
increases with increasing gas column density. This is unlikely to be due to
increasing dust extinction because the required dust to gas ratios are too
high. We suggest instead that this correlation arises because massive (i.e.
H-alpha producing) stars are formed preferentially in regions with high gas
density.
