We define a new distance measure - the resistor-average distance - between two probability distributions that is closely related to the Kullback-Leibler distance. While the Kullback-Leibler distance is asymmetric in the two distributions, the resistor-average distance is not. It arises from geometric considerations similar to those used to derive the Chernoff distance. Determining its relation to well-known distance measures reveals a newway to depict how commonly used distance measures relate to each other.