  We derive, in more general conditions, a recently introduced variance sum
rule (VSR) [I. Di Terlizzi et al., 2024 Science 383 971] involving variances of
displacement and force impulse for overdamped Langevin systems in a
nonequilibrium steady state (NESS). This formula allows visualising the effect
of nonequilibrium as a deviation of the sum of variances from normal diffusion
$2Dt$, with $D$ the diffusion constant and $t$ the time. From the VSR, we also
derive formulas for the entropy production rate $\sigma$ that, differently from
previous results, involve second-order time derivatives of position correlation
functions. This novel feature gives a criterion for discriminating strong
nonequilibrium regimes without measuring forces. We then apply and discuss our
results to three analytically solved models: a stochastic switching trap, a
Brownian vortex, and a Brownian gyrator. Finally, we compare the advantages and
limitations of known and novel formulas for $\sigma$ in an overdamped NESS.
