  We show a direct relationship between the variance and the differential
entropy for subclasses of symmetric and asymmetric unimodal distributions by
providing an upper bound on variance in terms of entropy power. Combining this
bound with the well-known entropy power lower bound on variance, we prove that
the variance of the appropriate subclasses of unimodal distributions can be
bounded below and above by the scaled entropy power. As differential entropy
decreases, the variance is sandwiched between two exponentially decreasing
functions in the differential entropy. This establishes that for the subclasses
of unimodal distributions, the differential entropy can be used as a surrogate
for concentration of the distribution.
