When using sheaves to model real-world problems -- e.g., topological signal processing or information integration -- we must face up to the challenges that real data provide. In a sheaf-theoretical context, measured data, called assignments, are represented by members of objects in the data category. Assignments need not be sections, and indeed, as measured data, almost always require statistical descriptions and tolerances. Uncertainty can exist between disparate measurements generally, and even on the same observable, due to uncertainty, noise, or sensor malfunctions.