Bounds on variance for symmetric unimodal distributions

We show a direct relationship between the variance and the differential entropy for the general class of symmetric 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 for the general class of symmetric unimodal distributions the variance 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 general class of symmetric unimodal distributions, the differential entropy can be used as a surrogate for concentration of the distribution.