Bounds on Variance for Symmetric Unimodal Distributions

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