 Observational entropy is a measure of entropy that can be used to compare systems in different states of equilibrium or out of equilibrium. It has been argued to provide a useful measure of out-of-equilibrium thermodynamic entropy. In this paper, the authors explore the mathematical properties of observational entropy from an information-theoretic perspective. They derive bounds on observational entropy for general cases, as well as bounds and identities related to sequential and post-processed measurements. Additionally, they introduce the concept of the coarse-grain state, which arises from the measurement statistics without requiring prior knowledge of the system's true state. This coarse-grain state is shown to provide upper and lower bounds on the difference between observational and von Neumann entropies. This article was authored by Francesco Buscemi, Joseph Schindler, and Dominic Afranek.