  Gladders et al. have recently suggested that the star formation histories
(SFHs) of individual galaxies are characterized by a log-normal function in
time, implying a slow decline rather than rapid quenching. We test their
conjecture on theoretical SFHs from the cosmological simulation Illustris and
on observationally inferred SFHs. While the log-normal form necessarily ignores
short-lived features such as starbursts, it fits the overall shape of the
majority of SFHs very well. In particular, 85% of the cumulative SFHs are
fitted to within a maximum error of 5% of the total stellar mass formed, and
99% to within 10%. The log-normal performs systematically better than the
commonly used delayed-tau model, and is superseded only by functions with more
than three free parameters. Poor fits are mostly found in galaxies that were
rapidly quenched after becoming satellites. We explore the log-normal parameter
space of normalization, peak time, and full width at half maximum, and find
that the simulated and observed samples occupy similar regions, though
Illustris predicts wider, later-forming SFHs on average. The ensemble of
log-normal fits correctly reproduces complex metrics such as the evolution of
Illustris galaxies across the star formation main sequence, but overpredicts
their quenching timescales. SFHs in Illustris are a diverse population not
determined by any one physical property of galaxies, but follow a tight
relation, where $\mathrm{width}\propto\mathrm{(peak\ time)}^{3/2}$. We show
that such a relation can be explained qualitatively (though not quantitatively)
by a close connection between the growth of dark matter halos and their
galaxies.
